Semi-neat rings: A generalization of neat ring structures

Abdallah A. Abukeshek; Andrew Rajah; Abdallah A. Abukeshek; Andrew Rajah

doi:10.3934/math.2025569

AIMS Mathematics

2025, Volume 10, Issue 5: 12619-12630. doi: 10.3934/math.2025569

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Research article

Semi-neat rings: A generalization of neat ring structures

Abdallah A. Abukeshek ^,,
Andrew Rajah

School of Mathematical Sciences, Universiti Sains Malaysia, 11800 Pulau Pinang, Malaysia

Received: 27 December 2024 Revised: 09 May 2025 Accepted: 19 May 2025 Published: 29 May 2025
MSC : 13A99, 13F99

This article introduces the concept of semi-neat rings, a generalization of neat rings, defined as rings whose nontrivial homomorphic images are semiclean. The study establishes fundamental properties of these rings, exploring their relationships with clean, neat, and semiclean rings. A classification of semi-neat FGC (Finitely Generated Cyclic) rings is provided, and the paper demonstrates that torch rings are semi-neat but not neat, illustrating the broader scope of the semi-neat concept. The article also investigates conditions under which the corner rings of semi-neat rings retained semi-neatness and explored extensions of the concept, such as weakly semi-neat rings. Constructed examples showed the existence of semi-neat rings that were not weakly neat, highlighting distinctions among related ring classes. The findings expanded the understanding of ring structures by demonstrating how semi-neat and weakly semi-neat rings were closed under homomorphic images and direct products. Applications to polynomial rings, triangular matrix rings, and torch rings revealed the utility of these generalized classes in algebra. The paper not only broadens the theoretical framework for neat and clean rings but also provides a foundation for further research on generalized ring properties and their implications in algebraic studies.
- clean ring,
- neat ring,
- weakly clean ring,
- semi-clean ring,
- semi-neat ring,
- weakly semi-neat ring
Citation: Abdallah A. Abukeshek, Andrew Rajah. Semi-neat rings: A generalization of neat ring structures[J]. AIMS Mathematics, 2025, 10(5): 12619-12630. doi: 10.3934/math.2025569

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Abstract

This article introduces the concept of semi-neat rings, a generalization of neat rings, defined as rings whose nontrivial homomorphic images are semiclean. The study establishes fundamental properties of these rings, exploring their relationships with clean, neat, and semiclean rings. A classification of semi-neat FGC (Finitely Generated Cyclic) rings is provided, and the paper demonstrates that torch rings are semi-neat but not neat, illustrating the broader scope of the semi-neat concept. The article also investigates conditions under which the corner rings of semi-neat rings retained semi-neatness and explored extensions of the concept, such as weakly semi-neat rings. Constructed examples showed the existence of semi-neat rings that were not weakly neat, highlighting distinctions among related ring classes. The findings expanded the understanding of ring structures by demonstrating how semi-neat and weakly semi-neat rings were closed under homomorphic images and direct products. Applications to polynomial rings, triangular matrix rings, and torch rings revealed the utility of these generalized classes in algebra. The paper not only broadens the theoretical framework for neat and clean rings but also provides a foundation for further research on generalized ring properties and their implications in algebraic studies.

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