Open AccessA NOTE ON COMMUTATIVITY OF PRIME NEAR RING WITH GENERALIZED β-DERIVATION
Author(s) -
Abdul Rauf Khan,
Khadija Mumtaz,
Muhammad Waqas
Publication year - 2021
Publication title -
matrix science mathematic
Language(s) - English
Resource type - Journals
eISSN - 2521-084X
pISSN - 2521-0831
DOI - 10.26480/msmk.01.2021.16.19
Subject(s) - prime (order theory) , homomorphism , mathematics , commutative ring , ring (chemistry) , commutative property , prime ring , pure mathematics , combinatorics , discrete mathematics , chemistry , organic chemistry
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: 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 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.