## Smarandache Non-Associative Rings (Google eBook)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

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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 |

139 | |

145 | |

### Common terms and phrases

associative rings Bernstein algebra called Characterize class of groupoids commutative loop ring commutative ring define Smarandache DEFINITION denoted field of characteristic finite loop following table Give an example group rings groupoid G groupoid ring RG idempotent inner commutative Jordan algebra Jordan identity Leibniz algebra Let RG Let Z2 Lie ideal Lie ring Ln(m loop algebra loop given loop ring RL mixed direct product modular lattice Moufang NA-ring nilpotent non-associative ring ofRG p-ring prime field Proof proper subset properties reader to prove regular elements right commutative right ideal S-groupoid S-ideals S-idempotents S-Jordan S-Lie ideal S-Lie ring S-loop S-normal S-pseudo ideals S-strong S-strongly S-subgroupoid S-subloop S-subring S-zero semi semi-ideal semigroup Smarandache analogue Smarandache Jordan algebras Smarandache Lie algebras Smarandache loop rings Smarandache pseudo Smarandache strongly SNA-ideal SNA-ring SNA-subring subalgebra subgroup subloop subring subspace vector space zero divisor ZLn(m

### 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.