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Ring homomorphism

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Title: Ring homomorphism  
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Ring homomorphism

In ring theory or abstract algebra, a ring homomorphism is a function between two rings which respects the structure.

More explicitly, if R and S are rings, then a ring homomorphism is a function f : RS such that[1][2][3][4][5][6]

  • f(a + b) = f(a) + f(b) for all a and b in R
  • f(ab) = f(a) f(b) for all a and b in R
  • f(1R) = 1S.

(Additive inverses and the additive identity are part of the structure too, but it is not necessary to require explicitly that they too are respected, because these conditions are consequences of the three conditions above. On the other hand, neglecting to include the condition f(1R) = 1S would cause several of the properties below to fail.)

If R and S are rngs (also known as pseudo-rings, or non-unital rings), then the natural notion[7] is that of a rng homomorphism, defined as above except without the third condition f(1R) = 1S. It is possible to have a rng homomorphism between (unital) rings that is not a ring homomorphism.

The composition of two ring homomorphisms is a ring homomorphism. It follows that the class (set theory) of all rings forms a category with ring homomorphisms as the morphisms (cf. the category of rings). In particular, one obtains the notions of ring endomorphism, ring isomorphism, and ring automorphism.

Properties

Let f : RS be a ring homomorphism. Then, directly from these definitions, one can deduce:

  • f(0R) = 0S.
  • f(−a) = −f(a) for all a in R.
  • For any unit element a in R, f(a) is a unit element such that f(a−1) = f(a)−1. In particular, f induces a group homomorphism from the (multiplicative) group of units of R to the (multiplicative) group of units of S (or of im(f)).
  • The image of f, denoted im(f), is a subring of S.
  • The kernel of f, defined as ker(f) = {a in R : f(a) = 0}, is an ideal in R. Every ideal in a commutative ring R arises from some ring homomorphism in this way.
  • The homomorphism f is injective if and only if ker(f) = {0}.
  • If f is bijective, then its inverse f−1 is also a ring homomorphism. In this case, f is called an isomorphism, and the rings R and S are called isomorphic. From the standpoint of ring theory, isomorphic rings cannot be distinguished.
  • If there exists a ring homomorphism f : RS then the characteristic of S divides the characteristic of R. This can sometimes be used to show that between certain rings R and S, no ring homomorphisms RS can exist.
  • If Rp is the smallest subring contained in R and Sp is the smallest subring contained in S, then every ring homomorphism f : RS induces a ring homomorphism fp : RpSp.
  • If R is a field and S is not the zero ring, then f is injective.
  • If both R and S are fields, then im(f) is a subfield of S, so S can be viewed as a field extension of R.
  • If R and S are commutative and P is a prime ideal of S then f−1(P) is a prime ideal of R.
  • If R and S are commutative and S is an integral domain, then ker(f) is a prime ideal of R.
  • If R and S are commutative, S is a field, and f is surjective, then ker(f) is a maximal ideal of R.
  • If f is surjective, P is prime (maximal) ideal in R and ker(f) ⊆ P, then f(P) is prime (maximal) ideal in S.

Moreover,

  • The composition of ring homomorphisms is a ring homomorphism.
  • The identity map is a ring homomorphism (but not the zero map).
  • Therefore, the class of all rings together with ring homomorphisms forms a category, the category of rings.
  • For every ring R, there is a unique ring homomorphism ZR. This says that the ring of integers is an initial object in the category of rings.
  • For every ring R, there is a unique ring homomorphism R → 0, where 0 denotes the zero ring (the ring whose only element is zero). This says that the zero ring is a terminal object in the category of rings.

Examples

  • The function f : ZZn, defined by f(a) = [a]n = a mod n is a surjective ring homomorphism with kernel nZ (see modular arithmetic).
  • The function f : Z6Z6 defined by f([a]6) = [4a]6 is a rng homomorphism (and rng endomorphism), with kernel 3Z6 and image 2Z6 (which is isomorphic to Z3).
  • There is no ring homomorphism ZnZ for n ≥ 1.
  • The complex conjugation CC is a ring homomorphism (in fact, an example of a ring automorphism.)
  • If R and S are rings, the zero function from R to S is a ring homomorphism if and only if S is the zero ring. (Otherwise it fails to map 1R to 1S.) On the other hand, the zero function is always a rng homomorphism.
  • If R[X] denotes the ring of all polynomials in the variable X with coefficients in the real numbers R, and C denotes the complex numbers, then the function f : R[X] → C defined by f(p) = p(i) (substitute the imaginary unit i for the variable X in the polynomial p) is a surjective ring homomorphism. The kernel of f consists of all polynomials in R[X] which are divisible by X2 + 1.
  • If f : RS is a ring homomorphism between the rings R and S, then f induces a ring homomorphism between the matrix rings Mn(R) → Mn(S).

The category of rings

Endomorphisms, isomorphisms, and automorphisms

  • A ring endomorphism is a ring homomorphism from a ring to itself.
  • A ring isomorphism is a ring homomorphism having a 2-sided inverse that is also a ring homomorphism. One can prove that a ring homomorphism is an isomorphism if and only if it is bijective as a function on the underlying sets. If there exists a ring isomorphism between two rings R and S, then R and S are called isomorphic. Isomorphic rings differ only by a relabeling of elements. Example: Up to isomorphism, there are four rings of order 4. (This means that there are four pairwise non-isomorphic rings of order 4 such that every other ring of order 4 is isomorphic to one of them.) On the other hand, up to isomorphism, there are eleven rngs of order 4.
  • A ring automorphism is a ring isomorphism from a ring to itself.

Monomorphisms and epimorphisms

Injective ring homomorphisms are identical to monomorphisms in the category of rings: If f : RS is a monomorphism that is not injective, then it sends some r1 and r2 to the same element of S. Consider the two maps g1 and g2 from Z[x] to R that map x to r1 and r2, respectively; fg1 and fg2 are identical, but since f is a monomorphism this is impossible.

However, surjective ring homomorphisms are vastly different from epimorphisms in the category of rings. For example, the inclusion ZQ is a ring epimorphism, but not a surjection. However, they are exactly the same as the strong epimorphisms.

Notes

  1. ^ Artin, p. 353
  2. ^ Atiyah and Macdonald, p. 2
  3. ^ Bourbaki, p. 102
  4. ^ Eisenbud, p. 12
  5. ^ Jacobson, p. 103
  6. ^ Lang, p. 88
  7. ^ Hazewinkel et al. (2004), p. 3. Warning: They use the word ring to mean rng.

References

See also

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