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AI Summary & Key Takeaways
AI Insight⚡ TL;DR(Too Long; Didn't Read)
Also, we prove that for a ring $R$, every $R$-module has $(C_{21})$ if and only if $R$ is a right t-semisimple ring.
📌 3 Key Takeaways
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Objective & Hypothesis:A module $M$ is called a $C_{21}$-module if, whenever $A$ and $B$ are submodules of $M$ with $A \cong B$, $A$ is nonsingular and $B$ is a direct summand of $M$, then $A$ is a direct summand of $M$.
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Key Findings & Data:Various examples of $C_{21}$-modules are presented.
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Conclusion & Impact:Also, we prove that for a ring $R$, every $R$-module has $(C_{21})$ if and only if $R$ is a right t-semisimple ring.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:$C_{21}$-module · $C_2$-module · nonsingular module
Abstract
A module $M$ is called a $C_{21}$-module if, whenever $A$ and $B$ are submodules of $M$ with $A \cong B$, $A$ is nonsingular and $B$ is a direct summand of $M$, then $A$ is a direct summand of $M$. Various examples of $C_{21}$-modules are presented. Some basic properties of these modules are investigated. It is shown that the class of rings $R$ over which every $C_{21}$-module is a $C_2$-module is exactly that of right SI-rings. Also, we prove that for a ring $R$, every $R$-module has $(C_{21})$ if and only if $R$ is a right t-semisimple ring.
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Cite This Article
Diallo, A. D., Dıop, P. C., Tribak, R. (2022). On a generalization of $C_2$-modules. Hacettepe Journal of Mathematics and Statistics, Volume: 51 Issue: 2. https://doi.org/10.15672/hujms.792212