On Strongly $Δ$-Clean Rings

TL;DR

This work introduces strongly $Δ$-clean rings, where each element splits as a sum of an idempotent and a $Δ(R)$-element that commute, with $Δ(R)$ the maximal subring of $J(R)$ invariant under units. It proves a sharp characterization: a ring is strongly $Δ$-clean exactly when it is strongly clean and $Δ(R) = \{ x \mid 1 - x \in U(R) \}$, and then derives extensive structural consequences across classical constructions, including Morita contexts, trivial extensions, triangular matrices, and group rings. The authors show fundamental links to Boolean quotients, local and 2-primal behavior, and establish non-strong-$Δ$-cleanness for matrix rings $M_n(R)$ with $n \ge 2$, while giving precise criteria for when various extensions preserve the property. Applications to group rings reveal that, for locally finite groups, $RG$ is strongly $Δ$-clean precisely when $R$ is and $G$ is a 2-group (under appropriate hypotheses), with notable restrictions when $G$ has odd-order finite subgroups. The results unify and extend existing clean-type ring theory, offering practical criteria and open problems for further study of $Δ$-related decompositions in algebraic structures.

Abstract

This study explores in-depth the structure and properties of the so-called {\it strongly $Δ$-clean rings}, that is a novel class of rings in which each ring element decomposes into a sum of a commuting idempotent and an element from the subset $Δ(R)$. Here, $Δ(R)$ stands for the extension of the Jacobson radical and is defined as the maximal subring of $J(R)$ invariant under the unit multiplication. We present a systematic framework for these rings by detailing their foundational characteristics and algebraic behavior under standard constructions, as well as we explore their key relationships with other well-established ring classes. Our findings demonstrate that all strongly $Δ$-clean rings are inherently strongly clean and $ΔU$, but under centrality constraints they refine the category of uniquely clean rings. Additionally, we derive criteria for the strong $Δ$-clean property in triangular matrix rings, their skew analogs, trivial extensions, and group rings. The analysis reveals deep ties to boolean rings, local rings, and quasi-duo rings by offering new structural insights in their algebraic characterization.

Theorems & Definitions (93)

• Definition 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• Lemma 2.4
• proof
• Example 2.5
• proof
• Lemma 2.6
• proof