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 p2=p∈comm2(a) such that a 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.