Rings Whose Clean and Nil-Clean Elements Have Some Clean-Like Properties
Peter Danchev, Arash Javan, Ahmad Moussavi
TL;DR
This work defines CSNC and NCUC rings and examines their place among existing clean/uniquely clean concepts, with abelian rings yielding notable equivalences between related classes. It provides comprehensive characterizations for CSNC rings, including the key criterion that for every clean element $a$, $a-a^2$ lies in ${\rm Nil}(R)$, and studies their behavior under corners, subrings, finite products, Morita contexts, and various matrix constructions, alongside implications for $J(R)$ and $R/J(R)$. It then demonstrates that NCUC rings are abelian by ensuring idempotents are uniquely clean, and shows that NCUC implies NCUNC, establishing a robust network of equivalent conditions tying abelianness to unique clean decompositions and related ring-structural properties. Overall, the results extend and unify prior work and provide practical criteria for identifying CSNC/NCUC rings across a range of algebraic constructions, including Morita-context and triangular matrix rings.
Abstract
We define two types of rings, namely the so-called CSNC and NCUC that are those rings whose clean elements are strongly nil-clean, respectively, whose nil-clean elements are uniquely clean. Our results obtained in this paper somewhat expand these obtained by Calugareanu-Zhou in Mediterr. J. Math. (2023) and by Cui-Danchev-Jin in Publ. Math. Debrecen (2024), respectively.