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arXiv · 2607.18125

Expanding Generalized Fine Rings

Ahmad Moussavi·Peter Danchev·Arash Javan·Omid Hasanzadeh
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Abstract

We introduce and study the so-called {\it generalized $\sqrt{J}$-fine rings}, where every element outside the Jacobson radical is the sum of a unit and an element from the set $\sqrt{J(R)} := \{ x \in R : x^{n} \in J(R) \text{ for some } n \ge 1 \}$. This commonly extends the notions of {\it fine} and {\it generalized fine rings} defined, respectively, by C\u{a}lug\u{a}reanu-Lam (J. Algebra \& Appl., 2016) and Zhou (J. Algebra \& Appl., 2022). Specifically, we prove that this class is closed under full matrix rings of any size, as well as we completely characterize when group rings over locally finite groups are generalized $\sqrt{J}$-fine. We also show that every such ring is 2-clean, thus properly placing it between generalized fine rings and 2-clean rings. Several examples are also provided to illustrate the complicated behavior of the introduced concept and its numerous boundaries.

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Ahmad Moussavi, Peter Danchev, Arash Javan, Omid Hasanzadeh. 2026-07-20. Expanding Generalized Fine Rings. https://arxiv.org/abs/2607.18125

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