a class of j-quasipolar rings

in this paper, we introduce a class of $j$-quasipolar rings. let $r$ be a ring with identity. an element $a$ of a ring $r$ is called {it weakly $j$-quasipolar} if there exists $p^2 = pin comm^2(a)$ such that $a + p$ or $a-p$ are contained in $j(r)$ and the ring $r$ is called {it weakly $j$-quasipolar} if every element of $r$ is weakly $j$-quasipolar. we give many characterizations and investigate general properties of weakly $j$-quasipolar rings. if $r$ is a weakly $j$-quasipolar ring, then we show that (1) $r/j(r)$ is weakly $j$-quasipolar, (2) $r/j(r)$ is commutative, (3) $r/j(r)$ is reduced. we use weakly $j$-quasipolar rings to obtain more results for $j$-quasipolar rings. we prove that the class of weakly $j$-quasipolar rings lies between the class of $j$-quasipolar rings and the class of quasipolar rings. among others it is shown that a ring $r$ is abelian weakly $j$-quasipolar if and only if $r$ is uniquely clean.