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AI Summary & Key Takeaways
AI Insight⚡ TL;DR(Too Long; Didn't Read)
Also, we investigate some conditions on ideals of $\Gamma$-ring that make $M$ to be commutative.
📌 3 Key Takeaways
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Objective & Hypothesis: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.
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Key Findings & Data: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.
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Conclusion & Impact:Also, we investigate some conditions on ideals of $\Gamma$-ring that make $M$ to be commutative.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:prime ring · commutativity · prime $\Gamma$-ring
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.
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Cite This Article
Kandamar, H., Arslan, O. (2020). On the commutativity conditions for rings and $\Gamma$-rings. Hacettepe Journal of Mathematics and Statistics, Volume: 49 Issue: 5. https://doi.org/10.15672/hujms.701857