On Strongly $\hat{I}”$-Clean Rings

Abstract

This study explores in-depth the structure and properties of the so-called {\it strongly $\hat{I}”$-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 $\hat{I}”(R)$. Here, $\hat{I}”(R)$ 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 $\hat{I}”$-clean rings are inherently strongly clean and $\hat{I}”U$, but under centrality constraints they refine the category of uniquely clean rings. Additionally, we derive criteria for the strong $\hat{I}”$-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.

21 pages