A CLASS OF J-QUASIPOLAR RINGS

Abstract. 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 weakly J-quasipolar if there exists p2 = p 2 comm2(a) such that a + p or a 􀀀 p are contained in J(R) and the ring R is called weakly J-quasipolar if every element of R is weakly J-quasipolar. We give many characterizations and investigate general proper- ties 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.