On Smarandache Rings II

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On Smarandache Rings II A K S CHANDRA SEKHAR RAO E-mail id: chandhrasekhar _ 6 @ yahoo.co.in Department of Mathematics D.N.R.College (Autonomous) Bhimavaram – 534 202, A.P, INDIA

Abstract In this paper we show that a commutative semisimple ring is always a Smarandache ring. We will also give a necessary and sufficient condition for group algebra to be a Smarandache ring. Examples are provided for justification.

Key Words: Group, ring, field, semisimple ring, group algebra, nilpotent element, Smarandache ring

1. Introduction

Smarandache notions which can be undoubtedly characterized as revolutionary mathematics, have the capacity of being utilized to analyze, study and introduce, naturally, the concepts of seven structures by means of extension or identification as a substructures. A particular case of Smarandache notions, an excellent means to study local properties in rings, is Smarandache ring [11].

The notion of semisimple ring is introduced in [7] and [8]. Semisimple rings have played prominent role in the development of structure theory of rings. There are several open questions concerning semisimplicity and group algebra of any group over a field.

Group algebras were introduced by G.Frobenius and I.Schur[4] in connection with the study of group representations, since studying the representations of a group G

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