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Smarandache Non-Associative Rings SMARANDACHE NON-ASSOCIATIVE RINGS W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technology, Madras Chennai – 600036, India e-mail: [email protected] web: http://mat.iitm.ac.in/~wbv Definition: 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 ⊂ 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 ∈ 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 ⊂ R, that is an associative ring (with respect to the same binary operations on R). 2 CONTENTS Preface 5 1. BASIC CONCEPTS 1 .1 B a sic s o f v e c to r sp a c e a n d b ilin e a r fo rm s 7 1 .2 S m a ra n d a c h e v e c to r sp a c e s 8 1 .3 B a sic d e fin itio n s o f o th e r a lg e b ra ic stru c tu re s 1 1 2. LOOP-RINGS AND SMARANDACHE LOOP-RINGS 2 .1 In tro d u c tio n o f lo o p s a n d S m a ra n d a c h e lo o p s 1 3 2 .2 P ro p e rtie s o f lo o p -rin g s a n d in tro d u c tio n to S m a ra n d a c h e lo o p -rin g s 2 0 2 .3 S m a ra n d a c h e e le m e n ts in lo o p rin g s 3 0 2 .4 S m a ra n d a c h e su b stru c tu re s in lo o p rin g s 4 0 2 .5 G e n e ra l p ro p e rtie s o f S m a ra n d a c h e lo o p rin g s a n d lo o p rin g s 53 3. GROUPOID RINGS AND SMARANDACHE GROUPOID RINGS 3 .1 G ro u p o id s a n d S m a ra n d a c h e G ro u p o id s 57 3 .2 G ro u p o id rin g s a n d S m a ra n d a c h e G ro u p o id rin g s 6 1 3 .3 S m a ra n d a c h e sp e c ia l e le m e n ts in G ro u p o id rin g s 6 9 3 .4 S m a ra n d a c h e su b stru c tu re s in G ro u p o id rin g s 76 3 .5 S p e c ia l p ro p e rtie s in G ro u p o id rin g s a n d S m a ra n d a c h e g ro u p o id rin g s 84 4. LIE ALGEBRAS AND SMARANDACHE LIE ALGEBRAS 4 .1 B a sic p ro p e rtie s o f L ie a lg e b ra s 9 1 4 .2 S m a ra n d a c h e L ie a lg e b ra s a n d its b a sic p ro p e rtie s 9 5 4 .3 S u b stru c tu re s in S m a ra n d a c h e L ie a lg e b ra s 9 9 4 .4 S p e c ia l p ro p e rtie s in S m a ra n d a c h e L ie a lg e b ra s 1 0 3 4 .5 S o m e n e w n o tio n s o n S m a ra n d a c h e L ie a lg e b ra s 1 0 6 3 5. JORDAN ALGEBRAS AND SMARANDACHE JORDAN ALGEBRAS 5.1 B a sic p ro p e rtie s o f J o rd a n a lg e b ra s 1 0 9 5.2 S m a ra n d a c h e J o rd a n a lg e b ra s a n d its b a sic p ro p e rtie s 1 1 3 6. SUGGESTIONS FOR FUTURE STUDY 6 .1 S m a ra n d a c h e g ro u p o id rin g s 1 1 9 6 .2 S m a ra n d a c h e lo o p rin g s 1 2 0 6 .3 L ie a lg e b ra s a n d S m a ra n d a c h e L ie a lg e b ra s 1 2 1 6 .4 S m a ra n d a c h e J o rd a n a lg e b ra s 1 2 2 6 .5 O th e r n o n -a sso c ia tiv e rin g s a n d a lg e b ra s a n d th e ir S m a ra n d a c h e a n a lo g u e 1 2 3 7. SUGGESTED PROBLEMS 1 2 7 References 1 3 9 Index 1 4 5 4 PREFACE 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 separate algebraic structures say a commutative ring with 1 (or a field) together with a loop or a groupoid or a vector space or a linear algebra. The two non-associative well-known algebras viz. Lie algebras and Jordan algebras are mainly built using a vector space over a field satisfying special identities called the Jacobi identity and Jordan identity respectively. Study of these algebras started as early as 1940s. Hence the study of non-associative algebras or even non-associative rings boils down to the study of properties of vector spaces or linear algebras over fields. But study of non-associative algebras using loops, that is loop algebras (or loop rings) was at its peak only in the 1980s. But till date there is no separate book on loop rings. So in this book we have given more importance to the study of loop rings and above all Smarandache loop rings. Further in my opinion the deeper study of loop rings will certainly branch off into at least five special type of algebras like Moufang algebras, Bol algebras, Bruck algebras, Alternative algebras and P- algebras. The author does not deny there are several research papers on these five special algebras but each of these algebras can be developed like Lie algebras and Jordan algebras. Another generalized class of non-associative algebras are from groupoid rings. The class of loop rings are strictly contained in the class of groupoid rings. Several important study in the direction of Smarandache notions has been done in this book. Groupoid rings are a very new concept, till date only very few research papers exist in the field of Groupoid rings. The notion of Smarandache Lie algebras and Smarandache Jordan algebras are introduced. There are several informative and exhaustive books on Lie algebras and Jordan algebras. Further as the study of these two algebras heavily exploits the concepts of vector spaces and matrix theory we do not venture to approach Smarandache Lie algebras or Smarandache Jordan algebras in this direction. But in this book we approach Smarandache Lie algebras and Smarandache Jordan algebras as a study in par with associative ringsfor such a study has not yet been approached by any researcher in this way. 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. 5 This book has seven chapters. First chapter is not only introductory but also devotes an entire section to introduce the basic concepts in Smarandache vector spaces. Chapter two studies about Smarandache loop rings and in chapter three, the Smarandache groupoid rings is introduced and analysed. Chapter four is devoted to the introduction of several new concepts in Lie algebras and Smarandache Lie algebras. Chapter five defines Smarandache Jordan algebras and gives several new notions so far not studied about in Jordan algebras and Smarandache Jordan algebras. Overall, we have introduced nearly 160 Smarandache notions relating to non- associative rings.
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