# Smarandache Non-Associative Rings (Google eBook)

Infinite Study, 2002 - Nonassociative rings - 149 pages
Generally, in any human field, a Smarandache Structure on a set A means a weak structure W on A such that there exists a proper subset B in A which is embedded with a stronger structure S. These types of structures occur in our everyday's life, that's why we study them in this book. Thus, as a particular case: A Non-associative ring is a non-empty set R together with two binary operations '+' and '.' such that (R, +) is an additive abelian group and (R, .) is a groupoid. For all a, b, c in R we have (a + b) . c = a . c + b . c and c . (a + b) = c . a + c . b. A Smarandache non-associative ring is a non-associative ring (R, +, .) which has a proper subset P in R, that is an associative ring (with respect to the same binary operations on R).

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### Contents

 BASIC CONCEPTS 7 12 Smarandache vector spaces 8 13 Basic definitions of other algebraic structures 11 LOOP RINGS AND SMARANDACHE LOOP RINGS 13 22 Properties of Loop Rings and Introduction to Smarandache loop rings 20 23 Smarandache Elements in Loop Rings 30 24 Smarandache substructures in loop rings 40 25 General properties of Smarandache loop rings and loop rings 53
 43 Substructures in Smarandache Lie algebras 99 44 Special Properties in Smarandache Lie Algebras 103 45 Some New Notions on Smarandache Lie Algebras 106 JORDAN ALGEBRAS AND SMARANDACHE JORDAN ALGEBRAS 109 52 Smarandache Jordan Algebras and its basic properties 113 SUGGESTIONS FOR FUTURE STUDY 119 62 Smarandache loop rings 120 63 Lie algebras and Smarandache Lie algebras 121

 GROUPOID RINGS AND SMARANDACHE GROUPOID RINGS 57 32 Groupoid rings and Smarandache groupoid rings 61 33 Smarandache special elements in Groupoid rings 69 34 Smarandache substructures in Groupoid rings 76 35 Special Properties in Groupoid rings and Smarandache Groupoid rings 84 LIE ALGEBRAS AND SMARANDACHE LIE ALGEBRAS 91 42 Smarandache Lie Algebras and its Basic Properties 95
 64 Smarandache Jordan Algebras 122 65 Other nonassociative rings and algebras and their Smarandache analogues 123 SUGGESTED PROBLEMS 127 REFERENCES 139 INDEX 145 Copyright

### Popular passages

Page 10 - For a fixed positive integer n, let ^„denote the set of all polynomials of degree less than or equal to n in F[x] which split over F.
Page 10 - Let V be a vector space over the field F and let T be a linear operator from V to V.
Page 7 - V. d. For each vector a in V there is a unique vector - a in V such that a + (-a.) = 0.
Page 10 - B generates a linear algebra over K then we call B a Smarandache strong basis (S-strong basis) for V.
Page 10 - Let V be a finite dimensional vector space over afield K. Let B = {vi , v2 , ..., v n } be a basis of V...
Page 7 - ... there is a unique vector 0 in V, called the zero vector, such that a + 0 = a for all a in V.
Page 5 - This book is distinct and different from other non-associative ring theory books as the properties of associative rings are incorporated and studied even in Lie rings and Jordan rings, an approach which is not traditional.