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AI Summary & Key Takeaways
AI Insight⚡ TL;DR(Too Long; Didn't Read)
We also prove that a $*$-ring $R$ is commutative, strongly nil $*$-clean and every primary ideal is maximal if and only if every element of $R$ is a projection.
📌 3 Key Takeaways
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Objective & Hypothesis:A $*$-ring $R$ is called {\em strongly nil $*$-clean} if every element of $R$ is the sum of a projection and a nilpotent element that commute with each other.
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Key Findings & Data:In this paper we investigate some properties of strongly nil $*$-rings and prove that $R$ is a strongly nil $*$-clean ring if and only if every idempotent in $R$ is a projection, $R$ is periodic, and $R/J(R)$ is Boolean.
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Conclusion & Impact:We also prove that a $*$-ring $R$ is commutative, strongly nil $*$-clean and every primary ideal is maximal if and only if every element of $R$ is a projection.
🔬Design:Araştırma Makalesi
📍Scope:Comprehensive Sample
⚙️Variables:Rings with involution · Boolean ring · Strongly nil *-clean ring
Abstract
A $*$-ring $R$ is called {\em strongly nil $*$-clean} if every element of $R$ is the sum of a projection and a nilpotent element that commute with each other. In this paper we investigate some properties of strongly nil $*$-rings and prove that $R$ is a strongly nil $*$-clean ring if and only if every idempotent in $R$ is a projection, $R$ is periodic, and $R/J(R)$ is Boolean. We also prove that a $*$-ring $R$ is commutative, strongly nil $*$-clean and every primary ideal is maximal if and only if every element of $R$ is a projection.
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Cite This Article
Harmanci, A., Chen, H., Ozcan, A. C. (2017). Strongly nil *-clean rings. Journal of Algebra Combinatorics Discrete Structures and Applications, Cilt: 4 Sayı: 2 (Special Issue: Noncommutative rings and their applications), 155–164. https://doi.org/10.13069/jacodesmath.284954