$\sqrt{\mathrm{\Delta }}$-Fine Rings

We introduce and study the so-termed {\it $\sqrt{\mathrm{\Delta }}$-fine rings}, a new class of rings that generalizes the classical {\it fine rings} introduced by Călugăreanu-Lam in J. Algebra & Appl. (2016) by requiring that every nonzero element $r\in R$ can be written as $r=u+a$, where $u$ is a unit and $a\in \sqrt{\mathrm{\Delta }(R)}$. We establish that every such ring is simple, every abelian $\sqrt{\mathrm{\Delta }}$-fine ring is indecomposable, and most notably, the matrix ring $M_n(R)$ over a $\sqrt{\mathrm{\Delta }}$-fine ring $R$ is again $\sqrt{\mathrm{\Delta }}$-fine for every $n\ge 1$. As a consequence, we characterize all semi-local $\sqrt{\mathrm{\Delta }}$-fine rings as those rings which are precisely the simple Artinian rings. We also examine group rings, providing conditions under which they are either $\sqrt{\mathrm{\Delta }}$-fine or generalized fine, where the latter class was introduced by Zhou in J. Algebra & Appl. (2022), and conclude our work with the difficult open question asking of whether each $\sqrt{\mathrm{\Delta }}$-fine ring is necessarily fine.