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arXiv · 2508.01286

Rings Whose Non-Units are Square-Nil Clean

Mina Doostalizadeh·Ahmad Moussavi·Peter Danchev
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Abstract

We consider in-depth and characterize in certain aspects the class of so-called {\it strongly NUS-nil clean rings}, that are those rings whose non-units are {\it square nil-clean} in the sense that they are a sum of a nilpotent and a square-idempotent that commutes with each other. This class of rings lies properly between the classes of strongly nil-clean rings and strongly clean rings. In fact, it is proved the valuable criterion that a ring $R$ is strongly NUS-nil clean if, and only if, $a^4-a^2\in Nil(R)$ for every $a\not\in U(R)$. In particular, a ring $R$ with only trivial idempotents is strongly NUS-nil clean if, and only if, $R$ is a local ring with nil Jacobson radical. Some special matrix constructions and group ring extensions will provide us with new sources of examples of NUS-nil clean rings.

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Mina Doostalizadeh, Ahmad Moussavi, Peter Danchev. 2025-08-02. Rings Whose Non-Units are Square-Nil Clean. https://arxiv.org/abs/2508.01286

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