Year: 2024
Type: article
Abstract: We consider and study those rings in which each nil-clean or clean element is uniquely nil-clean. We establish that those rings are abelian. More precisely, it is shown that the classes of abelian rings and the rings in which nil-clean elements are uniquely nil-clean do coincide. Moreover, we prove that the rings in which clean elements are uniquely nil-clean coincide with the subclass of abelian rings consisting of only unipotent units. In particular, we obtain in the semi-potent case a complete characterization only in terms of the former ring and its divisions. Likewise, some extension properties and group rings for such kinds of rings are also considered.
Authors: Jian Cui, Peter Danchev, Danya Jin
Language: English
Topic: Rings, Modules, and Algebras
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G1737173237, G6117409904, G9000067579
OpenAlex ID: W4403310438