Semi-neat rings: A generalization of neat ring structures

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.