Rings with $\Delta$-quasipolarity Property

This study provides a comprehensive investigation into the structure and properties of a novel class of rings known as $\Delta$-quasipolar rings, in which for every $a\in R$ there exisxt $p^2=p \in comm^2(a)$ such that $a+p \in \Delta(R)$. Here, $\Delta(R)$ is the largest Jacobson radical subring of $R$ which is closed with respect to multiplication by units of $R$. We have developed a general and comprehensive theoretical framework for $\Delta$-quasipolar rings. Within this framework, we have described the fundamental mathematical properties of these rings and examined their behavior under various classical algebraic structures, such as matrix rings and polynomial rings. Furthermore, we have investigated how this class of rings is related to other well-known ring types in mathematics, such as strongly $\Delta$-clean rings, uniquely clean rings, $J$-quasipolar rings and others.