Strongly weak idempotent nil -clean rings

Document Type : Research Paper

Authors

1 Department of Mathematics, College of Natural and Computational Science, Addis Ababa University, Addis Ababa, Ethiopia

2 Department of Computer Science and Systems Engineering, College of Engineering, Andhra University, Visakhapatnam, Andhra Pradesh, India

Abstract

We introduce the concept of strongly weak idempotent nil-clean rings which is a generalization of strongly weakly nil clean rings. We characterize strongly weak idempotent nil-clean rings in terms of the set of nilpotent elements, homomorphic images, and Jacobson radicals. We prove that a ring $R$ is strongly weak idempotent nil-clean if and only if for any $a\in R$, $a-a^3$ is nilpotent if and only if $Nil(R)$ forms an ideal and $R/{Nil(R)}$ is reduced weak idempotent nil-clean if and only if $R$ has no homomorphic image $\mathbb{Z}_3\oplus \mathbb{Z}_3$ and $a^2-a^4$ is nilpotent. Moreover, we prove that a strongly weak idempotent nil-clean ring $R$ with $2\in J(R)$ satisfies nil-involution property.

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References

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Journal of Algebra and Related Topics
Volume 13, Issue 1
July 2025
Pages 159-172

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