Smarandache Non-Associative Rings (Google eBook)

Front Cover
Infinite Study, 2002 - Nonassociative rings - 149 pages
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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

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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.

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