On idempotent-fine group rings

Omar Al-Mallah; Mohammed Abu-Saleem; Noômen Jarboui; Omar Al-Mallah; Mohammed Abu-Saleem; Noômen Jarboui

doi:10.3934/math.2026014

AIMS Mathematics

2026, Volume 11, Issue 1: 345-352. doi: 10.3934/math.2026014

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On idempotent-fine group rings

1.
Mathematics Department, Faculty of Science, Al-Balqa Applied University, Salt 19117, Jordan
2.
Department of Mathematics, Sultan Qaboos University, Al-Khod 123, Muscat, Oman

Received: 23 October 2025 Revised: 04 December 2025 Accepted: 09 December 2025 Published: 05 January 2026
MSC : 16U10, 16U80, 16U99, 16S34

Let $ A $ be an associative ring. A nonzero element $ t \in A $ is called fine if it can be written as $ t = n + v $, where $ v $ is a unit and $ n $ is a nilpotent element. A ring $ A $ is called an idempotent-fine ring if every nonzero idempotent in $ A $ is fine. Let $ A $ be a ring (respectively, an integral domain) of characteristic $ p^m $ for some prime $ p $ and positive integer $ m $, and let $ G $ be a locally finite nilpotent group (respectively, a locally finite group). We proved that $ A[G] $ is an idempotent-fine ring if and only if $ G $ is a $ p $-group. Moreover, if $ F $ is a field of characteristic $ p $ and $ F[G] $ is an idempotent-fine ring, then every nontrivial element $ g $ in the group $ G $ of finite order is a $ p $-element. Conversely, if G is a locally finite $ p $-group, then $ F[G] $ is an idempotent-fine ring.
- group ring,
- fine ring,
- idempotent-fine ring
Citation: Omar Al-Mallah, Mohammed Abu-Saleem, Noômen Jarboui. On idempotent-fine group rings[J]. AIMS Mathematics, 2026, 11(1): 345-352. doi: 10.3934/math.2026014

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Abstract

Let $ A $ be an associative ring. A nonzero element $ t \in A $ is called fine if it can be written as $ t = n + v $, where $ v $ is a unit and $ n $ is a nilpotent element. A ring $ A $ is called an idempotent-fine ring if every nonzero idempotent in $ A $ is fine. Let $ A $ be a ring (respectively, an integral domain) of characteristic $ p^m $ for some prime $ p $ and positive integer $ m $, and let $ G $ be a locally finite nilpotent group (respectively, a locally finite group). We proved that $ A[G] $ is an idempotent-fine ring if and only if $ G $ is a $ p $-group. Moreover, if $ F $ is a field of characteristic $ p $ and $ F[G] $ is an idempotent-fine ring, then every nontrivial element $ g $ in the group $ G $ of finite order is a $ p $-element. Conversely, if G is a locally finite $ p $-group, then $ F[G] $ is an idempotent-fine ring.

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