On the commutativity conditions for rings and $\Gamma$-rings

Authors : Hatice KANDAMAR, Okan ARSLAN

Pages : 1660-1666

Doi:10.15672/hujms.701857

View : 11 | Download : 5

Publication Date : 2020-10-06

Article Type : Research Paper

Abstract :Let $R$ be any ring. In this paper we observe the relation between the center of $R$-ring $R$ and the center of usual ring $R$ and then prove if the center of $R$-ring $R$ is nonzero, then $R$ is commutative as a ring. We also show that the common hypothesis \[a\alpha b\beta c=a\beta b\alpha c\: \text{for all}\: a,b,c\in M\:\text{and}\: \alpha,\beta\in\Gamma\] for a weak Nobusawa $\Gamma$-ring $M$ is sufficent for $M$ to be commutative. Also, we investigate some conditions on ideals of $\Gamma$-ring that make $M$ to be commutative.
Keywords : prime ring, commutativity, prime Gamma ring, ideal