On strongly Δ-clean rings

This study explores in depth the structure and properties of the so-called strongly[Formula: see text]-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 [Formula: see text] (see also [P. H. Tin and N. Q. Tien, A note on strongly clean elements, Asian-European J. Math. 18(11) (2025) 2550047.]). Here, [Formula: see text] stands for the extension of the Jacobson radical and is defined as the maximal subring of [Formula: see text] 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 [Formula: see text]-clean rings are inherently strongly clean and [Formula: see text], but under centrality constraints they refine the category of uniquely clean rings. Additionally, we derive criteria for the strong [Formula: see text]-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. Our results somewhat supply those established in part by Tin-Tien (cf. [P. H. Tin and N. Q. Tien, A note on strongly clean elements, Asian-European J. Math. 18(11) (2025) 2550047.]).