Matrix Science Mathematic (MSMK)

A NOTE ON COMMUTATIVITY OF PRIME NEAR RING WITH GENERALIZED β-DERIVATION

June 25, 2021 Posted by din In MSMK

ABSTRACT

A NOTE ON COMMUTATIVITY OF PRIME NEAR RING WITH GENERALIZED β-DERIVATION

Journal: Matrix Science Mathematic (MSMK)
Author: Abdul Rauf Khan, Khadija Mumtaz, Muhammad Mohsin Waqas

This is an open access article distributed under the Creative Commons Attribution License CC BY 4.0, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited

DOI: 10.26480/msmk.01.2021.16.19

In this paper, we prove commutativity of prime near rings by using the notion of β-derivations. Let M be a prime near ring. If there exist  and two sided generalized β-derivation G associated with the non-zero two sided β-derivation  on M, where  is a homomorphism, satisfying the following conditions:

  1. G([p_1,q_1 ])=〖p_1〗^(u_1 ) [β(p_1 ),β(q_1)]〖p_1〗^(v_1 ) ∀ p_1,q_1 ϵ M
  2. G([p_1,q_1 ])=〖p_1〗^(u_1 ) [β(p_1 ),β(q_1)]〖p_1〗^(v_1 ) ∀ p_1,q_1 ϵ M

Then M is a commutative ring.

Pages 16-19
Year 2021
Issue 1
Volume 5

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