- Hacettepe Journal of Mathematics and Statistics
- Vol: 49 Issue: 5
- On the commutativity conditions for rings and $\Gamma$-rings
On the commutativity conditions for rings and $\Gamma$-rings
Authors : Hatice Kandamar, Okan Arslan
Pages : 1660-1666
Doi:10.15672/hujms.701857
View : 13 | Download : 5
Publication Date : 2020-10-06
Article Type : Research
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