On Commutatativity of Primitive Rings with Some Identities

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International Journal of Research and Scientific Innovation (IJRSI) | Volume V, Issue IV, April 2018 | ISSN 2321–2705

On Commutatativity of Primitive Rings with Some Identities

B. Sridevi1 and Dr. D.V.Ramy Reddy2

IJRISS Call for paper

  1Assistant Professor of Mathematics, Ravindra College of Engineering for Women, Kurnool- 518002 A.P., India.
2Professor of Mathematics, AVR & SVR College of Engineering And Technology, Ayyaluru, Nandyal-518502, A.P., India

Abstract: – In this paper, we prove that some results on commutativity of primitive rings with some identities

Key Words: Commutative ring, Non associative primitive ring, Central

I. INTRODUCTION

A modification of Johnsen`s identity viz., (ab)2 = (ba)2  for all a, b in R for a non -associative ring R which has no element of additive order 2, is commutative was proved by R.N. Gupta [1].  R.D. Giri and others [2] generalized Gupta`s result by taking (ab)2 – (ba)2  Z(R).n this paper, we first study some commutativity theorems of non-associative primitive rings with some identities in the center. We show that some preliminary results that we need in the subsequent discussion and prove some commutativity theorems of non-associative rings and also  non-associative primitive ring with (ab)2 – ab   Z(R) or (ab)2 –ba  Z(R)    a , b in R is commutative. We also prove that if R is a non-associative primitive ring with identity (ab)2 – b(a2b)  Z(R) for all a, b in R is commutative. Also we prove that if R is an alternative prime ring with identity b (ab2) a – (ba2) b  Z(R) for all a, b in R, then R is commutative. Some commutativity theorems for certain non-associative rings, which are generalization for the results of Johnsen and others and R.N. Gupta, are proved in this paper. Johensen, Outcalt and Yaqub proved that if a non-associative ring R satisfy the identity (ab)2 = a2 b2 for all a, b in R, then R is commutative.   The generalization of this result proved by R.D. Giri and others states that if R is a non-associative primitive ring satisfies the identity (ab)2 – a2 b2  Z(R), where Z(R) denoted the center, then R is commutative.