  In this paper, we introduce a class of quasipolar rings which is a
generalization of $J$-quasipolar rings. Let $R$ be a ring with identity. An
element $a \in R$ is called {\it $\delta$-quasipolar} if there exists $p^2 =
p\in comm^2(a)$ such that $a + p$ is contained in $\delta(R)$, and the ring $R$
is called {\it $\delta$-quasipolar} if every element of $R$ is
$\delta$-quasipolar. We use $\delta$-quasipolar rings to extend some results of
$J$-quasipolar rings. Then some of the main results of $J$-quasipolar rings are
special cases of our results for this general setting. We give many
characterizations and investigate general properties of $\delta$-quasipolar
rings.
