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...Over the past 25 years, I have been immersed in research in Algebra and more particularly in ring theory. I embarked on writing this book on Smarandache rings (Srings) specially to motivate both ring theorists and Smarandache algebraists to develop and study several important and innovative properties about S-rings. ...

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...w. b. vasantha kandasamy smarandache rings american res... ...a 6 a 10 a 17 a 11 a 9 a 1 a 4 a 15 a 12 a 0 1 Smarandache Rings W. B. Vasantha Kandasamy Department of Mathematics Indian Inst... ...ce representation of the S-ideals of the Smarandache mixed direct product ring R = Z 3 × Z 12 × Z 7 . This is a major difference between a ring an... ...Z 12 × Z 7 . This is a major difference between a ring and a Smarandache ring. For, in a ring the lattice representation of ideals is always a modu... ...resentation of ideals is always a modular lattice but we see in case of S-rings the lattice representation of S-ideal need not in general be modular... ... 11 1.3 Lattices 14 1.4 Smarandache semigroups 19 2. Rings and its properties 2.1 Definition and examples 21 2.2 Specia... ...properties 2.1 Definition and examples 21 2.2 Special elements in rings 24 2.3 Substructures of a ring 26 2.4 Homomorphism and quot... ... 24 2.3 Substructures of a ring 26 2.4 Homomorphism and quotient rings 28 2.5 Special rings 30 2.6 Modules 34 2.7 Ring... ...s 34 2.7 Rings with chain conditions 35 3. Smarandache rings and its properties 3.1 Definition of Smarandache ring with example...

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...An associative ring is just realized or built using reals or complex; finite or infinite by defining two binary operations on it. But on the contrary when we want to define or study or even introduce a non-associative ring we need two separ...

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...w. b. vasantha kandasamy SMARANDACHE NON-ASSOCIATIVE RINGS AMERICAN RESEARCH PRESS REHOBOTH, NM 2002 1 ... ...ESEARCH PRESS REHOBOTH, NM 2002 1 Smarandache Non-associative rings W. B. Vasantha Kandasamy Department of Mathematics Indian Inst... ...ces 8 1.3 Basic definitions of other algebraic structures 11 2. LOOP-RINGS AND SMARANDACHE LOOP-RINGS 2.1 Introduction of loops and Smaranda... ...2.1 Introduction of loops and Smarandache loops 13 2.2 Properties of loop-rings and introduction to Smarandache loop-rings 20 2.3 Smarandache elemen... ...ntroduction to Smarandache loop-rings 20 2.3 Smarandache elements in loop rings 30 2.4 Smarandache substructures in loop rings 40 2.5 General prope... ...substructures in loop rings 40 2.5 General properties of Smarandache loop rings and loop rings 53 3. GROUPOID RINGS AND SMARANDACHE GROUPOID RIN... ...ROUPOID RINGS 3.1 Groupoids and Smarandache Groupoids 57 3.2 Groupoid rings and Smarandache Groupoid rings 61 3.3 Smarandache special elements i... ...marandache Groupoid rings 61 3.3 Smarandache special elements in Groupoid rings 69 3.4 Smarandache substructures in Groupoid rings 76 3.5 Special p... ...che substructures in Groupoid rings 76 3.5 Special properties in Groupoid rings and Smarandache groupoid rings 84 4. LIE ALGEBRAS AND SMARANDACH...

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...Near-rings are one of the generalized structures of rings. The study and research on near-rings is very systematic and continuous. Near-ring newsletters containing complete and updated bibliography on the subject are published per...

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...n 1941 by Higman [28]. He studied this relative to zero divisors in group rings and in fact proved if G is a two unique product group or a unique pr... ... G is a two unique product group or a unique product group then the group ring FG has no zero divisors where F is a field of characteristic zero. In... ..., negative if a < 1. This notion will find its importance in case of loop rings. DEFINITION 2.4.4: A loop L is called Hamiltonian if every subloo... ...Results in Mathematics) 12. Chein. O and Goodaire.E.D, Loops whose loop rings in characteristic 2 are alternative, Communications in Algebra, Vol.... ... 23. Goodaire.E.D and Parmenter.M.M, Semi simplicity of alternative loop rings, Acta. Math. Hung, Vol.50. 241-247 (1987). 24. Goodaire.E.D and R... ...ath., Vol. 34, No.3, 662-672 (1982). 25. Goodaire.E.D, Alternative loop rings, Publications Mathameticae (1-2), 31- 38 (1983). 26. Gratzer.G, La... ...lgebra, Wiley Eastern Limited, (1993). 28. Higman.G, The units of group rings, Proc. London Math. Soc., Vol.46, No.2, 231-248 (1940). 29. Ivan ... ...n. Vol.27, No.3, 499-518 (1986). 33. Kim.E.T, Idempotents in some group rings, Bull. Korean Math. Soc., Vol.24, No.2, 77-81 (1987). 34. Lang.S,... ..., No.4, 843-853 (1989). 56. Singh.S.V, On a new class of loops and loop rings, Ph.D. thesis IIT(Madras), guided by Vasantha.W.B., (1994). 57. S...

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...he Talmud states that if a respectable man marries, he doesn’t have to give a ring to his bride. On the contrary, his willingness to marry her is su... ...ccurs by examining one’s own attributes vs. those of the Creator, and prefer- ring those of the Creator. One doesn’t use one’s own attribute, such as... ...lds— , Atzi- lut, Beria, Yetzira and Assiya—are parallel. Try and imagine five rings with a joint center. Imagine the same picture in parallel, one u...

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...chehraa: an underwear, the specially designed breeches, (d) Karaa: a steel ring worn on the right - hand wrist and (e) Kirpaan: a small sword in bald...

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...asily, and draw conclusions for humans. At first glance, Nature seemed like a ring of egoists where only the fittest survive. This led researchers t...

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...m recent un roman de Danielle Steel, scriitoare american ă de succes, “The Ring”, ac ţiunea se petrecea în timpul şi dup ă cel de-al doilea r ăzboi ...

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...m recent un roman de Danielle Steel, scriitoare american ă de succes, “The Ring”, ac ţiunea se petrecea în timpul şi dup ă cel de-al doilea r ăzboi ...

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... forty Sefirot. It also implies that the Mochin are closed as though by a ring, which is the form of the Mem, so as to not receive Hochma. 47) Secon...

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...organ bisemigroups and bisemilattices was studied. It is said by Zoltan Esik that these bialgebraic structures like bigroupoids, bisemigroups, binear rings help in the construction of finite machines or finite automaton and semi automaton. The notion of non-associative bialgebraic structures was first introduced in the year 2002. The concept of bialgebraic structures which...

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...03 bigroups S-bigroups bigroupoids S-bigroupoids biloops S-biloops binear-rings S-binear-rings birings S-birings bisemigroups S-bisemigroups ... ...s bigroups S-bigroups bigroupoids S-bigroupoids biloops S-biloops binear-rings S-binear-rings birings S-birings bisemigroups S-bisemigroups ... ...s bigroups S-bigroups bigroupoids S-bigroupoids biloops S-biloops binear-rings S-binear-rings birings S-birings bisemigroups S-bisemigroups ... ...s bigroups S-bigroups bigroupoids S-bigroupoids biloops S-biloops binear-rings S-binear-rings birings S-birings bisemigroups S-bisemigroups ... ...s bigroups S-bigroups bigroupoids S-bigroupoids biloops S-biloops binear-rings S-binear-rings birings S-birings bisemigroups S-bisemigroups ... ...s bigroups S-bigroups bigroupoids S-bigroupoids biloops S-biloops binear-rings S-binear-rings birings S-birings bisemigroups S-bisemigroups ... ...s bigroups S-bigroups bigroupoids S-bigroupoids biloops S-biloops binear-rings S-binear-rings birings S-birings bisemigroups S-bisemigroups ... ...s bigroups S-bigroups bigroupoids S-bigroupoids biloops S-biloops binear-rings S-binear-rings birings S-birings bisemigroups S-bisemigroups ... ...s bigroups S-bigroups bigroupoids S-bigroupoids biloops S-biloops binear-rings S-binear-rings birings S-birings bisemigroups S-bisemigroups ...

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...rations. Semirings are algebraic structures with two binary operations enjoying several properties and it is the most generalized structure — for all rings and fields are semirings. The study of this concept is very meagre except for a very few research papers. Now, when we study the Smarandache semirings (S-semiring), we make the richer structure of semifield to be contai...

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... 1.1 Semigroups, groups and Smarandache semigroups 7 1.2 Lattices 11 1.3 Rings and fields 19 1.4 Vector spaces 22 1.5 Group rings and semigroup... ...oying several properties and it is the most generalized structure — for all rings and fields are semirings. The study of this concept is very meagre... ...ups, groups, lattices, Smarandache semigroups, fields, vector spaces, group rings and semigroup rings. We assume at the outset that the reader has a... ...w many Boolean algebras are there with four elements 0, 1, a and b? 1.3 Rings and Fields In this section we mainly introduce the concept of ri... ...ne to enable one to compare a field and a semifield. Second, to study group rings and semigroup rings. We do not give all the properties about field ... ...t the reader must have a good background of algebra. DEFINITION 1.3.1: A ring is a set R together with two binary operations + and • called addit... ... r • t and (r + s) • t = r • t + s • t (distributive law). We denote the ring by (R, +, •) or simply by R. In general the neutral element in (R,+)... ...rse of r ∈ R is –r. Instead of r • s we will denote it by rs. Clearly these rings are "associative rings". Let R be a ring, R is said to be commutat... ...or s = 0 for all r, s ∈ R then R is called integral. A commutative integral ring with identity is called an integral domain. If R \ {0} is a group...

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...ups and fuzzy subgroups 11 1.3 Fuzzy sub-bigroup of a group 32 1.4 Fuzzy rings and its properties 39 1.5 Fuzzy birings 64 1.6 Fuzzy fields and the... ...rties 79 1.7 Fuzzy semirings and their generalizations 84 1.8 Fuzzy near-rings and their properties 94 1.9 Fuzzy vector spaces and fuzzy bivector s... ...randache fuzzy biloops 275 3.4 Problems 282 4 4. SMARANDACHE FUZZY RINGS AND SMARANDACHE FUZZY NON-ASSOCIATIVE RINGS 4.1 Some fuzzy ... ...ector spaces and its properties 303 4.3 Smarandache fuzzy non-associative rings 309 4.4 Smarandache fuzzy birings and its properties 313 4.5 Proble... ... and its properties 354 5.5 Problems 363 6. SMARANDACHE FUZZY NEAR-RINGS AND ITS PROPERTIES 6.1 Smarandache fuzzy near-rings 369 6.2 Smara... ...arandache fuzzy near-rings 369 6.2 Smarandache non-associative fuzzy near-ring 381 6.3 Smarandache fuzzy binear-rings 389 6.4 Problems 398 7.... ...s have been working on concepts like fuzzy semigroup, fuzzy groups, fuzzy rings, fuzzy ideals, fuzzy semirings, fuzzy near-rings and so on. In t... ...both the books do not cover topics related to fuzzy semirings, fuzzy near-rings etc. so we have in this book, two parts: In Part 1 we have recalled ... ... fuzzy algebra which covers the notions of fuzzy semirings and fuzzy near-rings though there are several papers on these two concepts. This book...

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...1 N e w Y o r k C i t y Ades web magazine rockfeller center - ice skAting ring 202 N e w Y o r k C i t y Ades web magazine sAks & co 203 N e w Y o r... ...3 N e w Y o r k C i t y Ades web magazine rockfeller center - ice skAting ring , north -west corner 304 N e w Y o r k C i t y Ades web magazine rock... ...5 N e w Y o r k C i t y Ades web magazine rockfeller center - ice skAting ring 396 N e w Y o r k C i t y Ades web magazine tAxi 397 N e w Y o r k C ...

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...g, (1958). 2. W.B.Vasantha Kandasamy, On Ordered Groupoids and its Groupoid Rings, Jour. of Maths and Comp. Sci., Vol 9, 145-147, (1996). 31 ... ..., (1958). 2. W.B.Vasantha Kandasamy, On Ordered Groupoids and its Groupoids Rings, Jour. of Maths and Comp. Sci., Vol 9, 145-147, (1996). 3. W.... ... every groupoid Z n , {0} is not in general an ideal of Z n as in the case of rings. So whenever {0} happens to be an ideal we can consider it as...

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...s. We should understand that without experiencing that the changes are occur- ring outside us, we would not be able to make contact with the Creator...