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

Year 2020, Volume: 49 Issue: 5, 1660 - 1666, 06.10.2020

Hatice Kandamar Okan Arslan

https://doi.org/10.15672/hujms.701857

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

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