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
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.