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

#### Hatice KANDAMAR [1] , Okan ARSLAN [2]

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.

prime ring, commutativity, prime $\Gamma$-ring, ideal
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Primary Language en Mathematics Mathematics Orcid: 0000-0001-6391-5927Author: Hatice KANDAMAR Institution: ADNAN MENDERES UNIVERSITYCountry: Turkey Orcid: 0000-0002-1006-066XAuthor: Okan ARSLAN (Primary Author)Institution: ADNAN MENDERES UNIVERSITYCountry: Turkey Publication Date : October 6, 2020
 Bibtex @research article { hujms701857, journal = {Hacettepe Journal of Mathematics and Statistics}, issn = {2651-477X}, eissn = {2651-477X}, address = {}, publisher = {Hacettepe University}, year = {2020}, volume = {49}, pages = {1660 - 1666}, doi = {10.15672/hujms.701857}, title = {On the commutativity conditions for rings and \$\\Gamma\$-rings}, key = {cite}, author = {Kandamar, Hatice and Arslan, Okan} } APA Kandamar, H , Arslan, O . (2020). On the commutativity conditions for rings and $\Gamma$-rings . Hacettepe Journal of Mathematics and Statistics , 49 (5) , 1660-1666 . DOI: 10.15672/hujms.701857 MLA Kandamar, H , Arslan, O . "On the commutativity conditions for rings and $\Gamma$-rings" . Hacettepe Journal of Mathematics and Statistics 49 (2020 ): 1660-1666 Chicago Kandamar, H , Arslan, O . "On the commutativity conditions for rings and $\Gamma$-rings". Hacettepe Journal of Mathematics and Statistics 49 (2020 ): 1660-1666 RIS TY - JOUR T1 - On the commutativity conditions for rings and $\Gamma$-rings AU - Hatice Kandamar , Okan Arslan Y1 - 2020 PY - 2020 N1 - doi: 10.15672/hujms.701857 DO - 10.15672/hujms.701857 T2 - Hacettepe Journal of Mathematics and Statistics JF - Journal JO - JOR SP - 1660 EP - 1666 VL - 49 IS - 5 SN - 2651-477X-2651-477X M3 - doi: 10.15672/hujms.701857 UR - https://doi.org/10.15672/hujms.701857 Y2 - 2019 ER - EndNote %0 Hacettepe Journal of Mathematics and Statistics On the commutativity conditions for rings and $\Gamma$-rings %A Hatice Kandamar , Okan Arslan %T On the commutativity conditions for rings and $\Gamma$-rings %D 2020 %J Hacettepe Journal of Mathematics and Statistics %P 2651-477X-2651-477X %V 49 %N 5 %R doi: 10.15672/hujms.701857 %U 10.15672/hujms.701857 ISNAD Kandamar, Hatice , Arslan, Okan . "On the commutativity conditions for rings and $\Gamma$-rings". Hacettepe Journal of Mathematics and Statistics 49 / 5 (October 2020): 1660-1666 . https://doi.org/10.15672/hujms.701857 AMA Kandamar H , Arslan O . On the commutativity conditions for rings and $\Gamma$-rings. Hacettepe Journal of Mathematics and Statistics. 2020; 49(5): 1660-1666. Vancouver Kandamar H , Arslan O . On the commutativity conditions for rings and $\Gamma$-rings. Hacettepe Journal of Mathematics and Statistics. 2020; 49(5): 1660-1666.

Authors of the Article
[1]
[2]