Rings whose Non-Units are a Unit Multiple of an Element from $\sqrt{\Delta(R)}$

TL;DR

This paper introduces $U\sqrt{ riangle}$-rings, a new class where non-units are unit multiples of elements with powers in a special subring, and explores their properties and examples.

Contribution

It defines and studies $U\sqrt{ riangle}$-rings, establishing their basic properties, characterizations, and behavior under various ring constructions and extensions.

Findings

$U\sqrt{ riangle}$-rings are indecomposable and Dedekind-finite.

Polynomial and Laurent polynomial rings are never $U\sqrt{ riangle}$-rings.

Power series rings inherit the $U\sqrt{ riangle}$ property from their base rings.

Abstract

This paper introduces and studies a new class of rings called {\it $U\sqrt{\Delta}$-rings}. A ring $R$ is $U\sqrt{\Delta}$ if every non-unit element can be written as the product of a unit and an element from $\sqrt{\Delta(R)}$, where $\sqrt{\Delta(R)}$ consists of elements some power of which lies in the special subring $\Delta(R)$. We establish certain basic properties of these rings and, concretely, prove that they are simultaneously indecomposable and Dedekind-finite. We also show that the polynomial ring $R[x]$ and the Laurent polynomial ring $R[x, x^{-1}]$ are never $U\sqrt{\Delta}$-rings, while the power series ring $R[[x]]$ inherits this property from $R$. Likewise, for left (right) Artinian rings, the conditions of being a $U\sqrt{\Delta}$-ring and a $UN$-ring are equivalent, as well as these two conditions are preserved for the full matrix ring $M_n(R)$ of size $n\geq 1$ over $R$. In addition, for a commutative ring $R$, $M_n(R)$ is a $U\sqrt{\Delta}$-ring exactly when $R$ is local. Furthermore, we characterize when a group ring $RG$ is a $U\sqrt{\Delta}$-ring showing that, for a locally solvable group $G$, this occurs precisely when $R$ is a $U\sqrt{\Delta}$-ring and $G$ is a locally finite $p$-group for some prime $p \in J(R)$.