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# Unital algebra

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 Title: Unital algebra Author: World Heritage Encyclopedia Language: English Subject: Collection: Publisher: World Heritage Encyclopedia Publication Date:

### Unital algebra

In the mathematical study of abstract algebra, a ring is an algebraic structure generalizing the arithmetic operations of addition and multiplication. By means of this generalization, theorems from the algebra of arithmetic are applied to non-numerical objects like polynomials, series and functions.

Besides being studied in algebra, rings are used in many branches of mathematics, including geometry and mathematical analysis. The formal definition of rings is relatively recent (end of 19th century), and is an example of the tendency of modern mathematics to introduce, study, and manipulate abstract structures.

Briefly, a ring is an abelian group with a second binary operation that is distributive over the abelian group operation and is associative. The abelian group operation is called "addition" and the second binary operation is called "multiplication" in analogy with the integers. One familiar example of a ring is the set of integers. The integers are a commutative ring, since a times b is equal to b times a. The set of polynomials also forms a commutative ring. An example of a non-commutative ring is the ring of square matrices of the same size. Finally, a field (such as the real numbers) is a commutative ring in which one can do "division" by any nonzero element.

The ring theory consists of two main parts: the commutative ring theory, commonly known as commutative algebra, and the noncommutative ring theory. The former primarily concerns itself with problems and naturally occurring ideas in algebraic number theory and algebraic geometry. Important commutative rings include fields, polynomial rings, the coordinate rings of algebraic varieties and the rings of integers of number fields. On the other hand, the noncommutative theory is based on very different methods and, therefore, may not be viewed as a generalization of the commutative case. The major part of it is the structure theory: how a ring breaks up into simple pieces such as matrix rings. Many noncommutative rings come from analysis; e.g., operator algebras and rings of differential operators. Via group rings, the theory also found applications to the representation theory of groups. In geometry, the cohomology ring of a topological space constitutes an important geometric invariant of the space.

## Definition and illustration

The most familiar example of a ring is the set of all integers, Z, consisting of the numbers

. . . , −4, −3, −2, −1, 0, 1, 2, 3, 4, . . .

There are familiar properties for multiplication and addition of the integers. These properties serve as a model for the axioms for rings. A ring is a set R equipped with two binary operations + and · called addition and multiplication that map every pair of elements of R to a unique element of R. These operations must satisfy the following properties called ring axioms (the symbol ⋅ is often omitted and multiplication is just denoted by juxtaposition.), which must be true for all a, b, c in R:

1. (a + b) + c = a + (b + c) (+ is associative)
2. There is an element 0 in R such that 0 + a = a (0 is the zero element)
3. a + b = b + a (+ is commutative)
4. For each a in R there exists −a in R such that a + (−a) = (−a) + a = 0 (−a is the inverse element of a)
• Multiplication is associative:
5. (ab) ⋅ c = a ⋅ (bc)
6. a ⋅ (b + c) = (ab) + (ac) (left distributivity)
7. (b + c) ⋅ a = (ba) + (ca) (right distributivity)

For many authors, these seven axioms are all that are required in the definition of a ring (such a structure is also called pseudo-ring, or a rng). For others, the following additional axiom is also required:

• Multiplicative identity
8. There is an element 1 in R such that a ⋅ 1 = 1 ⋅ a = a

Rings which satisfy all eight of the above axioms are sometimes, for emphasis, referred to as unital rings (also called unitary rings, rings with unity, rings with identity or rings with 1). For example, the set of even integers satisfies the first seven axioms, but it does not have a multiplicative identity, and therefore does not satisfy the eighth axiom.

Regarding which convention should be used, Gardner and Wiegandt argue that if one requires all rings to have a unity, then some consequences include the lack of existence of infinite direct sums of rings, and the fact that proper direct summands of rings are not subrings. They conclude that "in many, maybe most, branches of ring theory the requirement of the existence of a unity element is not sensible, and therefore unacceptable." This article adopts the convention that, unless otherwise stated, a ring is assumed to be unital.

Although ring addition is commutative, so that a + b = b + a, ring multiplication is not required to be commutative; ab need not equal ba. Rings that also satisfy commutativity for multiplication (such as the ring of integers) are called commutative rings.

Some basic properties of a ring follow immediately from the axioms.

• The binomial formula holds for any commuting elements (i.e., $xy = yx$).

### Example: Integers modulo 4

Consider the set Z4 consisting of the numbers 0, 1, 2, 3 where addition and multiplication are defined as follows. To avoid possible confusions and to keep the usual notation for the arithmetic operations, we will over-line 0, 1, 2, 3 when considering them in Z4.

• (Addition) The sum $\overline\left\{x\right\} + \overline\left\{y\right\}$ in Z4 is the remainder of $x+y$ (as an integer) when divided by 4. For example, in Z4 we have $\overline\left\{2\right\} + \overline\left\{3\right\} = \overline\left\{1\right\}$ and $\overline\left\{3\right\} + \overline\left\{3\right\} = \overline\left\{2\right\}$
• (Multiplication) The product $\overline\left\{x\right\} \cdot \overline\left\{y\right\}$ in Z4 is the remainder of $x\cdot y$ (as an integer) when divided by 4. For example, in Z4 we have $\overline\left\{2\right\} \cdot \overline\left\{3\right\} = \overline\left\{2\right\}$ and $\overline\left\{3\right\} \cdot \overline\left\{3\right\} = \overline\left\{1\right\}.$

If x is an integer, the remainder of x when divided by 4 is an element of Z4, and this element is often denoted by "x mod 4", or sometimes $\overline\left\{x\right\}$, which is coherent with above notation. By checking each axiom, one verifies that Z4 is a ring under these operations. Each axiom follows from the fact that the integers form a ring, and converting the integers to Z4. The additive inverse of any $\overline\left\{x\right\}$ in Z4 is the remainder $\left(-x \mod 4\right) =\overline\left\{-4\right\}.$ In other words, we have $-\overline\left\{x\right\}=\overline\left\{-x\right\}.$ For example, in Z4, we have $-\overline\left\{3\right\}= \overline\left\{-3\right\} = \overline\left\{1\right\}.$

Once one has checked that the ring axioms hold, operations within the ring Z4 become easier to carry out. For example, to compute 3 ⋅ (3 − 1) + 1, one first computes the value within the full set of integers (which is 7), and then converts the result by finding the remainder after dividing by 4, which in this case is 3.

### Example: 2-by-2 matrices

Main article: 2 × 2 real matrices

Consider the set of 2-by-2 matrices, whose entries are real numbers. This set is written:

$\mathcal\left\{M\right\}_2\left(\mathbb\left\{R\right\}\right) = \left\\left\{ \begin\left\{pmatrix\right\} a & b \\ c & d \end\left\{pmatrix\right\} \bigg|\ a,b,c,d \in \mathbb\left\{R\right\} \right\\right\}$

One can check that with the operations of matrix addition and matrix multiplication, this set satisfies the above ring axioms. The element $\begin\left\{pmatrix\right\} 1 & 0 \\ 0 & 1 \end\left\{pmatrix\right\}$ is the multiplicative identity element of the ring. This ring is one of the simplest examples of a non-commutative ring. To see that it is not commutative, consider the following multiplications, which give two matrices A and B such that AB is different from BA:

$\begin\left\{pmatrix\right\} 0 & 1 \\ 1 & 0 \end\left\{pmatrix\right\}\begin\left\{pmatrix\right\} 0 & 1 \\ 0 & 0 \end\left\{pmatrix\right\} = \begin\left\{pmatrix\right\} 0 & 0 \\ 0 & 1 \end\left\{pmatrix\right\} \neq \begin\left\{pmatrix\right\} 1 & 0 \\ 0 & 0 \end\left\{pmatrix\right\} = \begin\left\{pmatrix\right\} 0 & 1 \\ 0 & 0 \end\left\{pmatrix\right\} \begin\left\{pmatrix\right\} 0 & 1 \\ 1 & 0 \end\left\{pmatrix\right\}$

One can generalize this construction by replacing the set of real numbers with any ring (not necessarily commutative), and instead of using 2-by-2 matrices, one can use square matrices of any fixed size; see matrix ring.

### Rings with extra structure

A ring may be viewed as an abelian group (by using the addition operation), with extra structure. In the same way, there are other mathematical objects which may be considered as rings with extra structure. For example:

• An associative algebra is a ring that is also a vector space over a field K. For instance, the set of n-by-n matrices over the real field R has dimension n2 as a real vector space.
• A ring R is a topological ring if its set of elements is given a topology which makes the addition map ( ) and the multiplication map ( ) to be both continuous as maps between topological spaces (where X × X inherits the product topology or any other product in the category). For example, n-by-n matrices over the real numbers could be given either the Euclidean topology, or the Zariski topology, and in either case one would obtain a topological ring.

## History

The study of rings originated from the theory of polynomial rings and the theory of algebraic integers. Furthermore, the appearance of hypercomplex numbers in the mid-19th century undercut the pre-eminence of fields in mathematical analysis.

In the 1880s Richard Dedekind introduced the concept of a ring, and the term Zahlring (Number ring) was coined by David Hilbert in 1892 and published in the article Die Theorie der algebraischen Zahlkörper, Jahresbericht der Deutschen Mathematiker Vereinigung, Vol. 4, 1897. In 19th-century German, the word "Ring" could mean "association", which is still used today in English in a limited sense (e.g., spy ring), so if that were the etymology then it would be similar to the way "group" entered mathematics by being a non-technical word for "collection of related things".

According to Harvey Cohn, Hilbert used the term for a specific ring that had the property of "circling directly back" to an element of itself. Specifically, in a ring of algebraic integers, all high powers of an algebraic integer can be written as an integral combination of a fixed set of lower powers, and thus the powers "cycle back". For instance, if a3 − 4a + 1 = 0 then a3 = 4a − 1, a4 = 4a2a, a5 = −a2 + 16a − 4, a6 = 16a2 − 8a + 1, a7 = −8a2 + 65a − 16, and so on; in general, an is going to be an integral linear combination of 1, a, and a2.

The first axiomatic definition of a ring was given by Adolf Fraenkel in an essay in Journal für die reine und angewandte Mathematik (A. L. Crelle), vol. 145, 1914. In 1921, Emmy Noether gave the first axiomatic foundation of the theory of commutative rings in her monumental paper Ideal Theory in Rings.

## Basic concepts

### Elements in a ring

A left zero-divisor is an element $a$ of a ring $R$ such that there exists a non-zero element $b$ of $R$ such that $ab = 0$. A right zero-divisor is defined similarly.

A nilpotent element is an element $a$ such that $a^n = 0$ for some $n > 0$. One example of a nilpotent element is a nilpotent matrix. A nilpotent element is necessarily a zero-divisor.

An idempotent $e$ is an element such that $e^2 = e$. One example of an idempotent element is a projection in linear algebra.

When an element $a$ has a multiplicative inverse, it is unique, is denoted by $a^\left\{-1\right\}$ and is called a unit. The set of units of a ring is a group under ring multiplication; this group is denoted by $U\left(R\right)$ or $R^*$. For example, if R is the ring of all square matrices of size n, $U\left(R\right)$ consists of the set of all invertible matrices of size n; called the general linear group.

### Subring

Main article: Subring

A subset S of R is said to be a subring if it can be regarded as a ring with the addition and the multiplication restricted from R to S. Equivalently, S is a subring if it is not empty, and for any x, y in S, $xy$, $x+y$ and $-x$ are in S. If all rings have been assumed, by convention, to have a multiplicative identity, then to be a subring one would also require S to share the same identity element as R. So if all rings have been assumed to have a multiplicative identity, then a proper ideal is not a subring.

For example, the ring Z of integers is a subring of the field of real numbers and also a subring of the ring of polynomials Z[X] (in both cases, Z contains 1, which is the multiplicative identity of the larger rings). On the other hand, the subset of even integers 2Z does not contain the identity element 1 and thus does not qualify as a subring.

The intersection of subrings is a subring. The smallest subring containing a given subset E of R is called a subring generated by E. Such a subring exists since it is the intersection of all subrings containing E.

For a ring R, the smallest subring containing 1 is called the characteristic subring of R. It can be obtained by adding copies of 1 and −1 together many times in any mixture. It is possible that $n\cdot 1=1+1+\ldots+1$ (n times) can be zero. If n is the smallest positive integer such that this occurs, then n is called the characteristic of R. In some rings, $n\cdot 1$ is never zero for any positive integer n, and those rings are said to have characteristic zero.

Given a ring R, let $\operatorname\left\{Z\right\}\left(R\right)$ denote the set of all elements x in R such that x commutes with every element in R: $xy = yx$ for any y in R. Then $\operatorname\left\{Z\right\}\left(R\right)$ is a subring of R; called the center of R. More generally, given a subset X of R, let S be the set of all elements in R that commute with every element in X. Then S is a subring of R, called the centralizer (or commutant) of X. The center is the centralizer of the entire ring R. Elements or subsets of the center are said to be central in R; they generate a subring of the center.

### Ideal

Main article: Ideal (ring theory)

The definition of an ideal in a ring is analogous to that of normal subgroup in a group. But, in actuality, it plays a role of an idealized generalization of an element in a ring; hence, the name "ideal". Like elements of rings, the study of ideals is central to structural understanding of a ring.

Let R be a ring. A subset I of R is then said to be a left ideal in R if $R I \subseteq I$. Here, $R I$ denotes the span of I over R; i.e., the set of finite sums

$r_1 x_1 + \cdots + r_n x_n, \quad r_i \in R, \quad x_i \in I.$

Similarly, I is said to be right ideal if $I R \subseteq I$. A subset I is said to be a two-sided ideal or simply ideal if it is both a left ideal and right ideal. A one-sided or two-sided ideal is then an additive subgroup of R. If E is a subset of R, then $R E$ is a left ideal, called the left ideal generated by E; it is the smallest left ideal containing E. Similarly, the two-sided ideal generated by E is the smallest two-sided ideal containing E, or, equivalently, $R E R.$

Given right (or left, or two-sided) ideals A and B of R, it is possible to show that the set intersection of A with B is an ideal of the same type as A and B. It is also possible to define a product of ideals such that AB is another ideal of the same sidedness as A and B. The sum of ideals A+B is also an ideal of the same type as A and B.

If x is in R, then $Rx$ and $xR$ are left ideals and right ideals, respectively; they are called the principal left ideals and right ideals generated by x. The principal ideal $RxR$ is written as $\left(x\right)$. For example, the set of all positive and negative multiples of 2 along with 0 form an ideal of the integers, and this ideal is generated by the integer 2. In fact, every ideal of the ring of integers is principal.

Like a group, a ring is said to be a simple if it is nonzero and it has no proper nonzero two-sided ideals. A commutative simple ring is precisely a field.

Rings are often studied with special conditions set upon their ideals. For example, a ring in which there is no strictly increasing infinite chain of left ideals is called a left Noetherian ring. A ring in which there is no strictly decreasing infinite chain of left ideals is called a left Artinian ring. It is a somewhat surprising fact that a left Artinian ring is left Noetherian (the Hopkins–Levitzki theorem). The integers, however, form a Noetherian ring which is not Artinian.

For commutative rings, the ideals generalize the classical notion of divisibility and decomposition of an integer into prime numbers in algebra. A proper ideal P of R is called a prime ideal if for any elements $x, y\in R$ we have that $xy \in P$ implies either $x \in P$ or $y\in P$. Equivalently, P is prime if for any ideals $I, J$ we have that $IJ \subseteq P$ implies either $I \subseteq P$ or $J \subseteq P.$ This later formulation illustrates the idea of ideals as generalizations of elements.

### Homomorphism

Main article: Ring homomorphism

A homomorphism from a ring (R, +, ·) to a ring (S, ‡, *) is a function f from R to S that preserves the ring operations; namely, such that, for all a, b in R the following identities hold:

• f(a + b) = f(a) ‡ f(b)
• f(a · b) = f(a) * f(b)
• f(1R) = 1S

If one is working with not necessarily unital rings, then the third condition is dropped.

A ring homomorphism is said to be an isomorphism if there exists an inverse homomorphism to f (i.e., a ring homomorphism which is an inverse function). Any bijective ring homomorphism is a ring isomorphism. Two rings $R, S$ are said to be isomorphic if there is an isomorphism between them and in that case one writes $R \simeq S$. A ring homomorphism between the same ring is called an endomorphism and an isomorphism between the same ring an automorphism.

Examples:

• The function that maps each integer x to its remainder modulo 4 (a number in {0, 1, 2, 3}) is a homomorphism from the ring Z to the quotient ring Z/4Z ("quotient ring" is defined below).
• If $u$ is a unit element in a ring R. Then $R \to R, x \mapsto uxu^\left\{-1\right\}$ is a ring homomorphism, called an inner automorphism of R.
• Let R be a commutative ring of prime characteristic p. Then $x \mapsto x^p$ is a ring endmorphism of R called the Frobenius homomorphism.
• The Galois group of a field extension $L/K$ is the set of all automorphisms of L whose restrictions to K are the identity.

Given a ring homomorphism $f:R \to S$, the set of all elements mapped to 0 by f is called the kernel of f. It is a two-sided ideal of R. The image of f, on the other hand, is not always an ideal, but it is always a subring of S.

To give a ring homomorphism from a commutative ring R to a ring A with image contained in the center of A is the same as to give a structure of an algebra over R to A (in particular gives a structure of A-module). Indeed, if f is such a map, one can define the scalar multiplication by R on A as $f\left(r\right)a$. Conversely, if A is an algebra over R, then $R \to A, r \mapsto r 1$ is such a ring homomorphism.

### Quotient ring

Main article: Quotient ring

The quotient ring of a ring, is analogous to the notion of a quotient group of a group. More formally, given a ring (R, +, · ) and a two-sided ideal I of (R, +, · ), the quotient ring (or factor ring) R/I is the set of cosets of I (with respect to the additive group of (R, +, · ); i.e. cosets with respect to (R, +)) together with the operations:

(a + I) + (b + I) = (a + b) + I and
(a + I)(b + I) = (ab) + I.

for every a, b in R.

Like the case of a quotient group, there is a canonical map $p: R \to R/I$ given by $x \mapsto x + I$. It is surjective and satisfies the universal property: if $f:R \to S$ is a ring homomorphism such that $f\left(I\right) = 0$, then there is a unique $\overline\left\{f\right\}: R/I \to S$ such that $f = \overline\left\{f\right\} \circ p$. In particular, taking I to be the kernel, one sees that the quotient ring $R / \operatorname\left\{ker\right\} f$ is isomorphic to the image of f; the fact known as the first isomorphism theorem. The last fact implies that actually any surjective ring homomorphism satisfies the universal property since the image of such a map is a quotient ring.

A subset of R and the quotient $R/I$ are related in the following way. A subset of R is called a system of representatives of $R/I$ if no two elements in the set belong to the same coset; i.e., each element in the set represents a unique coset. It is said to be complete if the restriction of $R \to R/I$ to it is surjective.

## Basic examples

Commutative rings:

• The motivating example is the ring of integers with the two operations of addition and multiplication. This is a commutative ring.
• The rational, real and complex numbers form commutative rings (in fact, they are even fields).
• The Gaussian integers form a ring, as do the Eisenstein integers. So does their generalization Kummer ring. cf. quadratic integers.
• In general, the set of all algebraic integers forms a ring. This follows for example from the fact that it is the integral closure of the ring of rational integers in the field of complex numbers. The rings in the previous example are subrings of this ring.
• The polynomial ring R[X] of polynomials over a ring R is also a ring.
• The set of formal power series R over a commutative ring R is a ring.
• If S is a set, then the power set of S becomes a ring if we define addition to be the symmetric difference of sets and multiplication to be intersection. This corresponds to a ring of sets and is an example of a Boolean ring.
• The set of all continuous real-valued functions defined on the real line forms a commutative ring. The operations are addition and multiplication of functions.
• Let R be the set of all continuous functions on the real line that vanish outside a bounded interval (an interval depends on a function). One can consider the following multiplication:
$fg\left(x\right) = \int_\left\{-\infty\right\}^\left\{\infty\right\} f\left(y\right)g\left(x-y\right)dy.$
R is then a ring but not unital: this is because if there were the multiplicative identity, it must be a Dirac delta function, which is not a continuous function.

Noncommutative rings:

• For any ring R and any natural number n, the set of all square n-by-n matrices with entries from R, forms a ring with matrix addition and matrix multiplication as operations. For n=1, this matrix ring is just (isomorphic to) R itself. For n>1, this matrix ring is an example of a noncommutative ring.
• If G is an abelian group, then the endomorphisms of G form a ring, the endomorphism ring End(G) of G. The operations in this ring are addition and composition of endomorphisms.
• If G is a group and R is a ring, the group ring of G over R is a free module over R having G as basis. Multiplication is defined by the rules that the elements of G commute with the elements of R and multiply together as they do in the group G.
• Many rings that appear in analysis are noncommutative. A basic example is a Banach algebra.

Non-example:

• The set of natural numbers N is not a ring, since (N, +) is not even a group (the elements are not all invertible with respect to addition). For instance, there is no natural number which can be added to 3 to get 0 as a result. There is a natural way to make it a ring by adding negative numbers to the set, thus obtaining the ring of integers. The natural numbers (including 0) form an algebraic structure known as a semiring (which has all of the properties of a ring except the additive inverse property).

## Constructions

### Direct product

Let R and S be rings. Then the product R × S can be equipped with the following natural ring structure:

• (r1, s1) + (r2, s2) = (r1 + r2, s1 + s2)
• (r1, s1) ⋅ (r2, s2) = (r1r2, s1s2)

for every r1, r2 in R and s1, s2 in S. The ring R × S with the above operations of addition and multiplication and the multiplicative identity $\left(1, 1\right)$ is called the direct product of R with S. The same construction also works for an arbitrary family of rings: if $R_i$ are rings indexed by a set I, then $\prod_\left\{i \in I\right\} R_i$ is a ring with componentwise addition and multiplication.

A "finite" direct product may also be viewed as a direct sum of ideals. Namey, let $R_i, 1 \le i \le n$ be rings, $R_i \to R = \prod R_i$ the inclusions with the images $\mathfrak\left\{a\right\}_i$ (in particular $\mathfrak\left\{a\right\}_i$ are rings though not subrings). Then $\mathfrak\left\{a\right\}_i$ are ideals of R and

$R = \mathfrak\left\{a\right\}_1 \oplus \cdots \oplus \mathfrak\left\{a\right\}_n, \quad \mathfrak\left\{a\right\}_i \mathfrak\left\{a\right\}_j = 0, i \ne j, \quad \mathfrak\left\{a\right\}_i^2 \subseteq \mathfrak\left\{a\right\}_i$

as a direct sum of abelian groups (because for abelian groups finite products are the same as direct sums). Clearly the direct sum of such ideals also defines a product of rings that is isomorphic to R. Equivalently, the above can be done through central idempotents. Assume R has the above decomposition. Then we can write

$1 = e_1 + \cdots + e_n, \quad e_i \in \mathfrak\left\{a\right\}_i.$

By the conditions on $\mathfrak\left\{a\right\}_i$, one has that $e_i$ are central idempotents and $e_i e_j = 0, i \ne j$ (orthogonal). Again, one can reverse the construction. Namely, if one is given a partition of 1 in orthogonal central idempotents, then let $\mathfrak\left\{a\right\}_i = R e_i$, which are two-sided ideals. If each $e_i$ is not a sum of orthogonal central idempotents, then their direct sum is isomorphic to R.

Let R be a commutative ring and $\mathfrak\left\{a\right\}_1, \cdots, \mathfrak\left\{a\right\}_n$ be ideals such that $\mathfrak\left\{a\right\}_i + \mathfrak\left\{a\right\}_j = \left(1\right)$ whenever $i \ne j$. Then the Chinese remainder theorem states that the map

$R \to R/ \mathfrak\left\{a\right\}_1 \times \cdots \times R/ \mathfrak\left\{a\right\}_n, \quad x \mapsto \left(x + \mathfrak\left\{a\right\}_1, \ldots , x + \mathfrak\left\{a\right\}_n\right)$

is a surjection with the kernel $\prod \mathfrak\left\{a\right\} _i = \cap \mathfrak\left\{a\right\}_i.$

An important application of an infinite direct product is the construction of a projective limit of rings, which carries over in verbatim from that for groups. Namely, suppose we're given a family of rings $R_i$, i running over positive integers, say, and ring homomorphisms $R_j \to R_i, j \ge i$ such that $R_i \to R_i$ are all the identities and $R_k \to R_j \to R_i$ is $R_k \to R_i$ whenever $k \ge j \ge i$. Then $\varprojlim R_i$ is the subring of $\prod R_i$ consisting of $\left(x_n\right)$ such that $x_j$ maps to $x_i$ under $R_j \to R_i, j \ge i$.

The notion of a subdirect product of rings generalizes a direct product of rings.

### Polynomial ring

Main article: Polynomial ring

Given a symbol t (i.e., a variable) and a commutative ring R, the set of polynomials

$R\left[t\right] = \left\\left\{ \sum_0^n a_j t^j \mid n \ge 0, a_j \in R \right\\right\}$

forms a commutative ring with the usual addition and multiplication, containing R as a subring. It is called the polynomial ring over R. More generally, the set $R\left[t_1, \ldots, t_n\right]$ of all polynomials in variables $t_1, \ldots, t_n$ forms a commutative ring, containing $R\left[t_i\right]$ as subrings.

If R is an integral domain, then $R\left[t\right]$ is also an integral domain; its field of fractions is the field of rational functions. If R is a noetherian ring, then $R\left[t\right]$ is a noetherian ring. If R is a unique factorization domain, then $R\left[t\right]$ is a unique factorization domain. Finally, R is a field if and only if $R\left[t\right]$ is a principal ideal domain.

Let $R \subseteq S$ be commutative rings. Given an element x of S, one can consider the ring homomorphism

$R\left[t\right] \to S, \quad f \mapsto f\left(x\right)$

(i.e., the substitution). If S=R[t] and x=t, then f(t)=f. Because of this, the polynomial f is often also denoted by $f\left(t\right)$. The image of the map $f \mapsto f\left(x\right)$ is denoted by $R\left[x\right]$; it is the same thing as the subring of S generated by R and x.

The substitution is a special case of the universal property of a polynomial ring. The property states: given a ring homomorphism $\phi: R \to S$ and an element x in S there exists a unique ring homomorphism $\overline\left\{\phi\right\}: R\left[t\right] \to S$ such that $\overline\left\{\phi\right\}\left(t\right) = x$ and $\overline\left\{\phi\right\}$ restricts to $\phi$. If a monic polynomial generates the kernel of $\overline\left\{\phi\right\}$, it is called the minimal polynomial of x over R. In a module-theoretic language, the universal property says that $R\left[x\right]$ is a free module over R with generators $1, x, x^2, \dots$.

To give an example, let S be the ring of all functions from R to itself; the addition and the multiplication are those of functions. Let x be the identity function. Each r in R defines a constant function, giving rise to the homomorphism $R \to S$. The universal property says that this map extends uniquely to

$R\left[t\right] \to S, \quad f \mapsto \overline\left\{f\right\}$

(t maps to x) where $\overline\left\{f\right\}$ is the polynomial function defined by f. The resulting map is injective if and only if R is infinite.

Given a non-constant monic polynomial f in $R\left[t\right]$, there exists a ring S containing R such that f is a product of linear factors in $S\left[t\right]$.

Let k be an algebraically closed field. The Hilbert's Nullstellensatz (theorem of zeros) states that there is a one-to-one correspondence between the set of all prime ideals in $k\left[t_1, \ldots, t_n\right]$ and the set of closed subvarieties of $k^n$. In particular, many local problems in algebraic geometry may be attacked through the study of the generators of an ideal in a polynomial ring. (cf. Gröbner basis.)

A quotient ring of the polynomial ring $R\left[t_1, \cdots, t_n\right]$ is said to be a finitely generated algebra over R or of finite type over R. The Noether normalization lemma says that any finitely generated commutative k-algebra R contains the polynomial ring with the coefficients in k over which R is finitely generated as a module. A polynomial ring is relatively well-understood and thus the theorem allows one to study a ring from the known facts about a polynomial ring.

There are some other related constructions. A formal power series ring $R\left[\!\left[t\right]\!\right]$ consists of formal power series

$\sum_0^\infty a_i t^i, \quad a_i \in R$

together with multiplication and addition that mimic those for convergent series. It contains $R\left[t\right]$ as a subring. The important advantage of a formal power series ring is that it is local (in fact, complete). One can also consider a polynomial ring in infinitely many variables $R\left[t_1, t_2, \dots\right]$: it is a union (i.e., direct limit) of $R\left[t_1, t_2, \dots, t_n\right]$ over all n. This ring is often used to furnish counterexamples.

Finally, there is a closely related notion: ring of polynomial functions on a vector space V. If V is a vector space over an infinite field, then, by choosing a basis, it may be identified with a polynomial ring.

### Endomorphism ring

Main article: Endomorphism ring

Let V be a module over a commutative ring R. Then $S = \operatorname\left\{End\right\}_R\left(V\right)$, the set of all R-linear maps, forms a ring with addition that is of function and multiplication that is of composition of functions. It is called the endomorphism ring of the module V.

The following construction is useful for an application to linear algebra (see also another example in the "Domain" section below.) Let $\phi: R \to S$ be given by $\phi\left(r\right)v = rv$ and $T: V \to V$ a linear map. Then $\phi\left(R\right)$ and T generate a commutative subring of S. By the universal property of a polynomial ring, $\phi$ uniquely extends to

$R\left[t\right] \to S, \quad f \mapsto f\left(T\right)$.

In particular, V acquires a structure of a module over $R\left[t\right]$; let $V_T$ denote the resulting module. This allows one to study T in terms of the module. For example, assuming R is a field, T is diagonalizable as a linear transformation if and only if $V_T$ is a semisimple module. For another example, $V_T, V_U$ are isomorphic as a module if and only if $T, U$ are similar; i.e., $T = H \circ U \circ H^\left\{-1\right\}$ for some isomorphism H.

Any ring with unity is the endomorphism ring of some object. (see the "category theoretic description" below.)

### Matrix ring

Main article: Matrix ring

Let R be a unital ring (not necessarily commutative). Then the set of all square matrices of size n forms a ring with the usual matrix addition and matrix multiplication. It is called the matrix ring and is denoted by $R_n$. As in linear algebra, a matrix ring may be interpreted as the set of linear mappings: $\operatorname\left\{End\right\}_R\left(R^n\right) \simeq R_n$. This will be a special case of the following: let $U$ be a left R-module. If $f: U^n \to U^n$ is a R-linear map, then f may be written as a matrix with entries $f_\left\{ij\right\}$ in $S = \operatorname\left\{End\right\}_R\left(U\right)$. Thus, we obtain a ring homomorphism

$\operatorname\left\{End\right\}_R\left(U^n\right) \to S_n, \quad f \mapsto \left(f_\left\{ij\right\}\right)$

which is clearly an isomorphism.

Let $e_\left\{ij\right\}$ be a matrix whose $\left(i, j\right)$-th entry is 1 and the other entries zero. If C is the centralizer in R of $e_\left\{ij\right\}$'s, then $R \simeq C_n$.

Any ring homomorhism $R \to S$ induces $R_n \to S_n$; in fact, any ring homomorphism $R_n \to S_n$ arises in this way.

Schur's lemma says that if U is a simple left R-module, then $\operatorname\left\{End\right\}_R\left(U\right)$ is a division ring. Let $\displaystyle U = \bigoplus_\left\{i = 1\right\}^r m_i U_i$ be a direct sum of R-modules where $U_i$ are simple modules and $mU_i$ means a direct sum of m copies of $U_i$. Then $\operatorname\left\{End\right\}_R\left(U\right)$ is isomorphic to a ring consisting of block diagonal matrices with matrices in $\operatorname\left\{End\right\}_R\left(U_i\right)_\left\{m_i\right\}$ in diagonal.

The ring R and its matrix ring $R_n$ are Morita equivalent. This implies for instance that there is a natural bijection between the set of two-sided ideals in R and the set of two-sided ideals in $R_n$.

Matrix rings often appear in the structure theorems in the noncommutative ring theory. (cf. Artin–Wedderburn theorem)

### Localization

The localization generalizes the construction of the field of fractions of an integral domain to an arbitrary commutative ring and modules. When the localization is applied to a ring with a prime ideal, it results in a local ring, a ring with the only one maximal ideal; hence, the term "localization".

Formally, let M be a right module over a ring R and S a multiplicatively closed subset of the center of R (i.e., xy in S for any x, y in S and S has 1 but not 0). Let $S^\left\{-1\right\} M$ be the set of equivalence classes of pairs $\left(m, s\right), m \in M, s \in S$ denoted by $\displaystyle \left\{m \over s\right\}$: two pairs $\left(m, s\right)$, $\left(n, t\right)$ are equivalent if there is a $u \in S$ such that

$\left(mt - ns\right)u = 0.$

We make $S^\left\{-1\right\} M$ a right R-module by defining the addition and scalar multiplication exactly like we defined the field of fractions. If M happens to be a ring (in particular R), we can also define the multiplication to make $S^\left\{-1\right\} M$ a ring. $S^\left\{-1\right\}M$ can also be viewed as $S^\left\{-1\right\} R$-module in the obvious way.

There is the canonical map $M \to S^\left\{-1\right\}M, \, m \mapsto m / 1$. Its kernel consists of elements m such that $ms = 0$ for some s in S.

If S happens to be the complement of a prime ideal $\mathfrak\left\{p\right\}$, then we write $M_\mathfrak\left\{p\right\}$ for $S^\left\{-1\right\} M$ and call it the localization of M at $\mathfrak\left\{p\right\}$. The field of fractions of an integral domain R is the localization of R at the prime ideal zero. If $\mathfrak\left\{p\right\}$ is a prime ideal of a commutative ring R, then the field of fractions of $R/\mathfrak\left\{p\right\}$ is the same as the residue field of the local ring $R_\mathfrak\left\{p\right\}$ and is denoted by $k\left(\mathfrak\left\{p\right\}\right)$.

Perhaps the most important properties of localization are the following: when R is a commutative ring and S multiplicatively closed subset

• $\mathfrak\left\{p\right\} \mapsto S^\left\{-1\right\}\mathfrak\left\{p\right\}$ is a bijection between the set of all prime ideals in R disjoint from S and the set of all prime ideals in $S^\left\{-1\right\}R$.
• $S^\left\{-1\right\}R = \varinjlim R\left[f^\left\{-1\right\}\right]$, f running over elements in S with partial ordering given by divisibility. Here, $R\left[f^\left\{-1\right\}\right]$ is the localization with respect to $\\left\{1, f, f^2, \cdots \\right\}$.
• The localization is exact:
$0 \to S^\left\{-1\right\}M\text{'} \to S^\left\{-1\right\}M \to S^\left\{-1\right\}M$ \to 0 is exact over $S^\left\{-1\right\} R$ whenever $0 \to M\text{'} \to M \to M$ \to 0 is exact over R.
• Conversely, if $0 \to M\text{'}_\mathfrak\left\{m\right\} \to M_\mathfrak\left\{m\right\} \to M$_\mathfrak{m} \to 0 is exact for any maximal ideal $\mathfrak\left\{m\right\}$, then $0 \to M\text{'} \to M \to M$ \to 0 is exact.
• A remark: localization is no help in proving a global existence. One instance of this is that if two modules are isomorphic at all prime ideals, it does not follow that they are isomorphic. (One way to explain this is that the localization allows one to view a module as a sheaf over prime ideals and a sheaf is inherently a local notion.)

### Completion

Given a noetherian commutative local ring $\left(R, \mathfrak\left\{m\right\}\right)$, the projective limit $\overline\left\{R\right\} = \varprojlim R/\mathfrak\left\{m\right\}^n$ is called the completion of R. The basic example is a completion of the ring Z of rational integers at a prime number. Given a prime number p, the localization Z(p) at the prime ideal generated by p is a local ring. Its competition is then called the ring of p-adic integers. Similarly, the formal power series ring $R\left[\!\left[t\right]\!\right]$ can be viewed as the completion of the localization of the polynomial ring $R\left[t\right]$ at $\left(t\right)$.

Like localization, there is a canonical map $R \to \overline\left\{R\right\}, x \mapsto \left(x \text\left\{ mod \right\} \mathfrak\left\{m\right\}^n\right)$. It is injective since the kernel $\cap_k \mathfrak\left\{m\right\}^k$ vanishes by the Krull's intersection theorem.

The completion is also carried out through the means of valuations. Given a field F, a valuation on F is a multiplicative group homomorphism from F to the field of real numbers such that only zero maps to zero and the triangle inequality holds. The basic example, besides absolute value, is a p-adic valuation on Q given by

$|0|_p = 0, |n|_p=p^\left\{-v_p\left(n\right)\right\}$

where the integer $v_p\left(n\right)$ is the number of p that appear in the unique factorization of a nonzero integer n into prime numbers. This valuation is nonarchimedean in the sense that $|x + y| = |x|$ whenever $|x| > |y|$. More generally, the same definition applies to a Dedekind domain since in such a ring there is a unique factorization of a principal ideal into prime ideals. Another example is this. Let F be a finite extension of Q. Then each homomorphism $\sigma: F \to \mathbb\left\{C\right\}$ determines a valuation $x \mapsto |\sigma\left(x\right)|$. This one is not nonarchimedean.

A valuation defines the distance function in the obvious way and this makes F a metric space. One can apply the procedure of the completion of a metric space and obtain the complete metric space $\overline\left\{F\right\}$ in which F is a dense subspace. The field operations (addition, subtraction, multiplication and inversion) on F uniquely extend to F and $\overline\left\{F\right\}$ becomes a field in the unique way as well. If the valuation is nonarchimedean, then the set

$O = \\left\{ x \in \overline\left\{F\right\} | |x| \le 1 \\right\}$

is an open compact subring of $\overline\left\{F\right\}$. It is a local ring called the ring of integers of the completion of $F$.

The complication between integral closure and completion is a starting point of modern commutative ring theory. Pathological examples found by Nagata led to the introduction of such a ring as excellent ring.

### Group ring

This construction allows one to study a group using the ring theory. Let G be a group and A a commutative ring. The group ring $A\left[G\right]$ of G over A is then the set of all functions $f: G \to A$ such that $f\left(s\right) = 0$ for all but finitely many s in G with addition and multiplication defined as follows. Let $R = A\left[G\right]$ and make it an abelian group with the ordinary addition of functions. The multiplication on it is given by convolution:

$\left(f*g\right)\left(t\right) = \sum_\left\{s \in G\right\} f\left(s\right)g\left(s^\left\{-1\right\}t\right)$.

This is a finite sum and is therefore well-defined. Also, the function $f*g$ belongs to $R$. One then checks that the addition and the multiplication satisfy the ring axioms. R has the multiplicative identity $\delta_1$ where $\delta_t\left(t\right) = 1, \delta_t\left(s\right) = 0$ for all $s \ne t$. (cf. Kronecker delta.) $\delta_t, \, t \in G,$ form an A-basis of R. Explicitly, for any f in R, there is the expansion

$f = \sum_\left\{s \in G\right\} f\left(s\right) \delta_s.$

Finally, essentially the same construction is possible for a unital semigroup instead of a group except the multiplication is given by:

$\left(f*g\right)\left(t\right) = \sum_\left\{uv = t\right\} f\left(u\right)g\left(v\right)$.

The resulting ring is called a semigroup ring. For example, $A\left[\mathbb\left\{N\right\}_0\right]$ is a polynomial ring of one variable over A.

### Tensor product

Let A, B be algebras over a commutative ring R. Then the tensor product of R-modules $A \otimes_R B$ is a R-module. We can turn it to a ring by extending linearly $\left(x \otimes u\right) \left(y \otimes v\right) = xy \otimes uv$. For example, if R' is an R-algebra, then $R\text{'} \otimes_R R\left[t\right] \simeq R\text{'}\left[t\right]$ and $R\left[t\right] \otimes_R R\left[t\right] \simeq R\left[t_1, t_2\right]$. To give a nontrivial example, let $\epsilon, t$ be indeterminates and view $R, R\left[\epsilon\right]$ as algebras over $R\left[t\right]$ via $t \mapsto 0, t \mapsto \epsilon^2$. One can show:

$R\left[\epsilon\right] \otimes_\left\{R\left[t\right]\right\} R \simeq R\left[\epsilon\right]/\left(\epsilon^2\right)$

(The ring on the right is called the ring of dual numbers.)

If A is a k-algebra, then $A_n \simeq A \otimes_k k_n$. We also have: for algebras A, A' over k and their subalgebras B, B', resp.,

$C_\left\{A \otimes A\text{'}\right\}\left(B \otimes B\text{'}\right) = C_A\left(B\right) \otimes C_\left\{A\text{'}\right\}\left(B\text{'}\right)$

where $C_A\left(B\right)$ refers to the centralizer of B in A. In particular, the center of $A \otimes B$ is the tensor product of the centers of A and B.

### Rings with generators and relations

The most general way to construct a ring is by specifying generators and relations. Let F be a free ring (i.e., free algebra over the integers) with the set X of symbols; i.e., F consists of polynomials with integral coefficients in noncommuting variables that are elements of X. A free ring satisfies the universal property that one expects on a free object; that is, any function from the set X to a ring R factors through F so that $F \to R$ is the unique ring homomorphism. Just as in the group case, a ring is a quotient of a free ring.

Now, we can impose relations among symbols in X by taking a quotient. Explicitly, if E is a subset of F, then the quotient ring of F by the ideal generated by E is called the ring with generators X and relations E. If we used a ring, say, A as a base ring instead of Z, then the resulting ring will be over A. For example, if $E = \\left\{ xy - yx \mid x, y \in X \\right\}$, then the resulting ring will be the usual polynomial ring with coefficients in A in variables that are elements of X (It is also the same thing as the symmetric algebra over A with symbols X.)

## Special kinds of rings

### Domains

A unital nontrivial ring with no nonzero zero-divisors is called a domain. A commutative domain is called an integral domain. The most important integral domains are principal ideals domains, PID for short, and fields. A principal ideal domain is a domain in which every ideal is principal. An important class of domains that contain a PID is a unique factorization domain, an integral domain in which every nonunit element is a product of prime elements. (an element x is prime if $\left(x\right)$ is a prime ideal.) The fundamental question in algebraic number theory is on the extent to which the ring of integers (not necessarily rational integers) fails to be a PID.

Among theorems concerning a PID, the most important one is the structure theorem for finitely generated modules over a principal ideal domain. The theorem may be illustrated by the following application to linear algebra. Let V be a finite-dimensional vector space over a field k and $f: V \to V$ a linear map with minimal polynomial q. Then, since $k\left[t\right]$ is a unique factorization domain, q factors into powers of distinct irreducible polynomials (i.e., prime elements):

$q = p_1^\left\{e_1\right\} ... p_s^\left\{e_s\right\}.$

Letting $t \cdot v = f\left(v\right)$, we make V a k[t]-module. The structure theorem then says that $V = \bigoplus V_i$ as k[t]-module where each $V_i$ is isomorphic to a direct sum of submodules $W$ isomorphic to $k\left[t\right]/\left(p_i^\left\{k_j\right\}\right)$. Now, if $p_i\left(t\right) = t - \lambda_i$, then such a $W$ has a basis in which the restriction of f is represented by a Jordan matrix, Thus, if, say, k is algebraically closed, then $p_i$ are of the form $t - \lambda_i$ and the above decomposition corresponds to the Jordan canonical form of f.

Any nonzero subring of a field is necessarily an integral domain. The converse is also true: an integral domain is always a subring of its field of fractions. This only partially generalizes to a noncommutative setting.

In algebraic geometry, UFD's arise because of smoothness. More precisely, a point in a variety (over a perfect field) is smooth if the local ring at the point is a regular local ring. A regular local ring is a UFD.

The following is a chain of class inclusions that describes the relationship between rings, domains and fields:

### Division ring

A division ring is a ring such that every non-zero element is a unit. A commutative division ring is a field. A prominent example of a division ring that is not a field is the ring of quaternions. Any centralizer in a division ring is also a division ring. In particular, the center of a division ring is a field. It turned out that every finite domain (in particular finite division ring) is a field; in particular commutative (the Wedderburn's little theorem).

The study of conjugacy classes figures prominently in the classical theory of division rings. Cartan famously asked the following question: given a division ring D and a proper sub-division-ring S that is not contained in the center, does each inner automorphism of D restrict to an automorphism of S? The answer is negative: this is the Cartan–Brauer–Hua theorem.

### Semisimple rings

A ring is called a semisimple ring if it is semisimple as a left module (or right module) over itself. A ring is called a semiprimitive ring if its Jacobson radical is zero. (The Jacobson radical is the intersection of all maximal left ideals.) A ring is semisimple if and only if it is artinian and is semiprimitive.

An algebra over a field k is artinian if and only if it has finite dimension. Thus, a semisimple algebra over a field is necessarily finite-dimensional, while a simple algebra may have infinite-dimension; e.g., the ring of differential operators.

Any module over a semisimple ring is semisimple. (Proof: any free module over a semisimple ring is clearly semisimple and any module is a quotient of a free module.)

Examples of semisimple rings:

• A matrix ring over a division ring is semisimple (actually simple).
• The group ring $k\left[G\right]$ of a finite group G over a field k is semisimple if the characteristic of k does not divide the order of G. (Maschke's theorem)
• The Weyl algebra (over a field) is a simple ring; it is not semisimple since it has infinite dimension and thus not artinian.
• Clifford algebras are semisimple.

Semisimplicity is closely related to separability. An algebra A over a field k is said to be separable if the base extension $A \otimes_k F$ is semisimple for any field extension $F/k$. If A happens to be a field, then this is equivalent to the usual definition in field theory (cf. separable extension.)

### Central simple algebra and Brauer group

For a field k, a k-algebra is central if its center is k and is simple if it is a simple ring. Since the center of a simple k-algebra is a field, any simple k-algebra is a central simple algebra over its center. In this section, a central simple algebra is assumed to have finite dimension. Also, we mostly fix the base field; thus, an algebra refers to a k-algebra. The matrix ring of size n over a ring R will be denoted by $R_n$.

The Skolem–Noether theorem states any automorphism of a central simple algebra is inner.

Two central simple algebras A and B are said to be similar if there are integers n and m such that $A \otimes_k k_n \approx B \otimes_k k_m$. Since $k_n \otimes_k k_m \simeq k_\left\{nm\right\}$, the similarity is an equivalence relation. The similarity classes $\left[A\right]$ with the multiplication $\left[A\right]\left[B\right] = \left[A \otimes_k B\right]$ form an abelian group called the Brauer group of k and is denoted by $\operatorname\left\{Br\right\}\left(k\right)$. By the Artin–Wedderburn theorem, a central simple algebra is the matrix ring of a division ring; thus, each similarity class is represented by a unique division ring.

For example, $\operatorname\left\{Br\right\}\left(k\right)$ is trivial if k is a finite field or an algebraically closed field (more generally quasi-algebraically closed field; cf. Tsen's theorem). $\operatorname\left\{Br\right\}\left(\mathbb\left\{R\right\}\right)$ has order 2 (a special case of the theorem of Frobenius). Finally, if k is a nonarchimedean local field (e.g., $\mathbb\left\{Q\right\}_p$), then $\operatorname\left\{Br\right\}\left(k\right) = \mathbb\left\{Q\right\}/\mathbb\left\{Z\right\}$ through the invariant map.

Now, if F is a field extension of k, then the base extension $- \otimes_k F$ induces $\operatorname\left\{Br\right\}\left(k\right) \to \operatorname\left\{Br\right\}\left(F\right)$. Its kernel is denoted by $\operatorname\left\{Br\right\}\left(F/k\right)$. It consists of $\left[A\right]$ such that $A \otimes_k F$ is a matrix ring over F (i.e., A is split by F.) If the extension is finite and Galois, then $\operatorname\left\{Br\right\}\left(F/k\right)$ is canonically isomorphic to $H^2\left(\operatorname\left\{Gal\right\}\left(F/k\right), k^*\right)$.

Azumaya algebras generalize the notion of central simple algebras to a commutative local ring.

## Structures and invariants of rings

### Dimension of a commutative ring

The Krull dimension of a commutative ring R is the supremum of the lengths n of all the increasing chains of prime ideals $\mathfrak\left\{p\right\}_0 \subsetneq \mathfrak\left\{p\right\}_1 \subsetneq \cdots \subsetneq \mathfrak\left\{p\right\}_n$. For example, the polynomial ring $k\left[t_1, \cdots, t_n\right]$ over a field k has dimension n. The fundamental theorem in the dimension theory states the following numbers coincide for a noetherian local ring $\left(R, \mathfrak\left\{m\right\}\right)$:

• The Krull dimension of R.
• The minimum number of the generators of the $\mathfrak\left\{m\right\}$-primary ideals.
• The dimension of the graded ring $\operatorname\left\{gr\right\}_\left\{\mathfrak\left\{m\right\}\right\}\left(R\right) = \oplus_\left\{k \ge 0\right\} \mathfrak\left\{m\right\}^k/\left\{\mathfrak\left\{m\right\}^\left\{k+1\right\}\right\}$ (equivalently, one plus the degree of its Hilbert polynomial).

A commutative ring R is said to be catenary if any pair of prime ideals $\mathfrak\left\{p\right\} \subset \mathfrak\left\{p\right\}\text{'}$ can be extended to a chain of prime ideals $\mathfrak\left\{p\right\} = \mathfrak\left\{p\right\}_0 \subsetneq \cdots \subsetneq \mathfrak\left\{p\right\}_n = \mathfrak\left\{p\right\}\text{'}$ of same finite length such that there is no prime ideal that is strictly contained in two consecutive terms. Practically all noetherian rings that appear in application are catenary. If $\left(R, \mathfrak\left\{m\right\}\right)$ is a catenary local integral domain, then, by definition,

$\operatorname\left\{dim\right\}R = \operatorname\left\{ht\right\}\mathfrak\left\{p\right\} + \operatorname\left\{dim\right\}R/\mathfrak\left\{p\right\}$

where $\operatorname\left\{ht\right\}\mathfrak\left\{p\right\} = \operatorname\left\{dim\right\}R_\left\{\mathfrak\left\{p\right\}\right\}$ is the height of $\mathfrak\left\{p\right\}$. It is a deep theorem of Ratliff that the converse is also true.

If R is an integral domain that is a finitely generated k-algebra, then its dimension is the transcendence degree of its field of fractions over k. If S is an integral extension of a commutative ring R, then S and R have the same dimension.

Closely related concepts are those of depth and global dimension. In general, if R is a noetherian local ring, then the depth of R is less than or equal to the dimension of R. When the equality holds, R is called a Cohen–Macaulay ring. A regular local ring is an example of a Cohen–Macaulay ring. It is a theorem of Serre that R is a regular local ring if and only if it has finite global dimension and in that case the global dimension is the Krull dimension of R. The significance of this is that a global dimension is a homological notion.

### Morita equivalence

Main article: Morita equivalence

Two rings R, S are said to be Morita equivalent if the category of left modules over R is equivalent to the category of left modules over S. In fact, two commutative rings which are Morita equivalent must be isomorphic, so the notion does not add anything new to the category of commutative rings. However, commutative rings can be Morita equivalent to noncommutative rings, and so Morita equivalence can be seen as a strictly coarser equivalence than isomorphism classes of rings. Morita equivalence is especially important in algebraic topology and functional analysis.

### Finitely generated projective module over a ring and Picard group

Let R be a commutative ring and $\mathbf\left\{P\right\}\left(R\right)$ the set of isomorphism classes of finitely generated projective modules over R; let also $\mathbf\left\{P\right\}_n\left(R\right)$ subsets consisting of those with constant rank n. (The rank of a module M is the continuous function $\operatorname\left\{Spec\right\}R \to \mathbb\left\{Z\right\}, \, \mathfrak\left\{p\right\} \mapsto \dim M \otimes_R k\left(\mathfrak\left\{p\right\}\right)$.) $\mathbf\left\{P\right\}_1\left(R\right)$ is usually denoted by Pic(R). It is an abelian group called the Picard group of R. If R is an integral domain with the field of fractions F of R, then there is an exact sequence of groups:

$1 \to R^* \to F^* \overset\left\{f \mapsto fR\right\}\to \operatorname\left\{Cart\right\}\left(R\right) \to \operatorname\left\{Pic\right\}\left(R\right) \to 1$

where $\operatorname\left\{Cart\right\}\left(R\right)$ is the set of fractional ideals of R. If R is a regular domain (i.e., regular at any prime ideal), then Pic(R) is precisely the divisor class group of R.

For example, if R is a principal ideal domain, then Pic(R) vanishes. In algebraic number theory, R will be taken to be the ring of integers, which is Dedekind and thus regular. It follows that Pic(R) is a finite group (finiteness of class number) that measures the deviation of the ring of integers from being a PID.

One can also consider the group completion of $\mathbf\left\{P\right\}\left(R\right)$; this results in a commutative ring K0(R). Note that K0(R) = K0(S) if two commutative rings R, S are Morita equivalent.

### Structure of noncommutative rings

Main article: Noncommutative ring

The structure of a noncommutative ring is more complicated than that of a commutative ring. For example, there exist rings which contain non-trivial proper left or right ideals, but are still simple; that is contain no non-trivial proper (two-sided). Various invariants exist for commutative rings, whereas invariants of noncommutative rings are difficult to find. As an example, the nilradical of a ring, the set of all nilpotent elements, need not be an ideal unless the ring is commutative. Specifically, the set of all nilpotent elements in the ring of all n x n matrices over a division ring never forms an ideal, irrespective of the division ring chosen. There are, however, analogues of the nilradical defined for noncommutative rings, that coincide with the nilradical when commutativity is assumed.

The concept of the Jacobson radical of a ring; that is, the intersection of all right/left annihilators of simple right/left modules over a ring, is one example. The fact that the Jacobson radical can be viewed as the intersection of all maximal right/left ideals in the ring, shows how the internal structure of the ring is reflected by its modules. It is also a fact that the intersection of all maximal right ideals in a ring is the same as the intersection of all maximal left ideals in the ring, in the context of all rings; whether commutative or noncommutative.

Noncommutative rings serve as an active area of research due to their ubiquity in mathematics. For instance, the ring of n-by-n matrices over a field is noncommutative despite its natural occurrence in geometry, physics and many parts of mathematics. More generally, endomorphism rings of abelian groups are rarely commutative, the simplest example being the endomorphism ring of the Klein four-group.

One of the best known noncommutative rings is the division ring of quaternions.

## Applications

### The ring of integers of a number field

Main article: Ring of integers

### The coordinate ring of an algebraic variety

If X is an affine algebraic variety, then the set of all regular functions on X forms a ring called the coordinate ring of X. For a projective variety, there is an analogus ring called the homogeneous coordinate ring. Those rings are essentially the same things as varieties: they correspond in essentially a unique way. This may be seen via either Hilbert's Nullstellensatz or scheme-theoretic constructions (i.e., Spec and Proj).

### Ring of invariants

A basic (and perhaps the most fundamental) question in the classical invariant theory is to find and study polynomials in the polynomial ring $k\left[V\right]$ that are invariant under the action of a finite group (or more generally reductive) G on V. The main example is the ring of symmetric polynomials: symmetric polynomials are polynomials that are invariant under permutation of variable. The fundamental theorem of symmetric polynomials states that this ring is $R\left[\sigma_1, \ldots, \sigma_n\right]$ where $\sigma_i$ are elementary symmetric polynomials.

## Some examples of the ubiquity of rings

Many different kinds of mathematical objects can be fruitfully analyzed in terms of some associated ring.

### Cohomology ring of a topological space

To any topological space X one can associate its integral cohomology ring

$H^*\left(X,\mathbb\left\{Z\right\}\right) = \bigoplus_\left\{i=0\right\}^\left\{\infty\right\} H^i\left(X,\mathbb\left\{Z\right\}\right),$

a graded ring. There are also homology groups $H_i\left(X,\mathbb\left\{Z\right\}\right)$ of a space, and indeed these were defined first, as a useful tool for distinguishing between certain pairs of topological spaces, like the spheres and tori, for which the methods of point-set topology are not well-suited. Cohomology groups were later defined in terms of homology groups in a way which is roughly analogous to the dual of a vector space. To know each individual integral homology group is essentially the same as knowing each individual integral cohomology group, because of the universal coefficient theorem. However, the advantage of the cohomology groups is that there is a natural product, which is analogous to the observation that one can multiply pointwise a k-multilinear form and an l-multilinear form to get a (k + l)-multilinear form.

The ring structure in cohomology provides the foundation for characteristic classes of fiber bundles, intersection theory on manifolds and algebraic varieties, Schubert calculus and much more.

### Burnside ring of a group

To any group is associated its Burnside ring which uses a ring to describe the various ways the group can act on a finite set. The Burnside ring's additive group is the free abelian group whose basis are the transitive actions of the group and whose addition is the disjoint union of the action. Expressing an action in terms of the basis is decomposing an action into its transitive constituents. The multiplication is easily expressed in terms of the representation ring: the multiplication in the Burnside ring is formed by writing the tensor product of two permutation modules as a permutation module. The ring structure allows a formal way of subtracting one action from another. Since the Burnside ring is contained as a finite index subring of the representation ring, one can pass easily from one to the other by extending the coefficients from integers to the rational numbers.

### Representation ring of a group ring

To any group ring or Hopf algebra is associated its representation ring or "Green ring". The representation ring's additive group is the free abelian group whose basis are the indecomposable modules and whose addition corresponds to the direct sum. Expressing a module in terms of the basis is finding an indecomposable decomposition of the module. The multiplication is the tensor product. When the algebra is semisimple, the representation ring is just the character ring from character theory, which is more or less the Grothendieck group given a ring structure.

### Function field of an irreducible algebraic variety

To any irreducible algebraic variety is associated its function field. The points of an algebraic variety correspond to valuation rings contained in the function field and containing the coordinate ring. The study of algebraic geometry makes heavy use of commutative algebra to study geometric concepts in terms of ring-theoretic properties. Birational geometry studies maps between the subrings of the function field.

### Face ring of a simplicial complex

Every simplicial complex has an associated face ring, also called its Stanley–Reisner ring. This ring reflects many of the combinatorial properties of the simplicial complex, so it is of particular interest in algebraic combinatorics. In particular, the algebraic geometry of the Stanley–Reisner ring was used to characterize the numbers of faces in each dimension of simplicial polytopes.

## Category theoretical description

Main article: Category of rings

Every ring can be thought of as a monoid in Ab, the category of abelian groups (thought of as a monoidal category under the tensor product of $\left\{\mathbb Z\right\}$-modules). The monoid action of a ring R on an abelian group is simply an R-module. Essentially, an R-module is a generalization of the notion of a vector space – where rather than a vector space over a field, one has a "vector space over a ring".

Let (A, +) be an abelian group and let End(A) be its endomorphism ring (see above). Note that, essentially, End(A) is the set of all morphisms of A, where if f is in End(A), and g is in End(A), the following rules may be used to compute f + g and f · g:

• (f + g)(x) = f(x) + g(x)
• (f · g)(x) = f(g(x))

where + as in f(x) + g(x) is addition in A, and function composition is denoted from right to left. Therefore, associated to any abelian group, is a ring. Conversely, given any ring, (R, +, · ), (R, +) is an abelian group. Furthermore, for every r in R, right (or left) multiplication by r gives rise to a morphism of (R, +), by right (or left) distributivity. Let A = (R, +). Consider those endomorphisms of A, that "factor through" right (or left) multiplication of R. In other words, let EndR(A) be the set of all morphisms m of A, having the property that m(r · x) = r · m(x). It was seen that every r in R gives rise to a morphism of A: right multiplication by r. It is in fact true that this association of any element of R, to a morphism of A, as a function from R to EndR(A), is an isomorphism of rings. In this sense, therefore, any ring can be viewed as the endomorphism ring of some abelian X-group (by X-group, it is meant a group with X being its set of operators). In essence, the most general form of a ring, is the endomorphism group of some abelian X-group.

## Generalization

Ring theorists have considered the generalization of a ring by weakening or dropping some of ring axioms.

### Nonassociative rings

A nonassociative ring is an algebraic structure that satisfies all of the ring axioms but the associativity. A notable example is a Lie algebra. There exists some structure theory for such algebra that generalizes the analog results for Lie algebras and associative algebras.

### Semirings

A semiring is obtained by replacing the underlying abelian group structure of a ring with a semigroup structure.