Research Information System
FILOMAT, vol.38, no.25, pp.8877-8891, 2024 (SCI-Expanded)
Idempotent elements, invertible elements and quasinilpotents are some important tools to study structures of rings. By using these kinds of elements, we study a class of rings, called strongly quasipolar rings, which is a subclass of that of quasipolar rings. Let R be a ring with identity. An element a is an element of R is said to be strongly quasipolar if there exists p(2 )= p is an element of comm(2)(a) such that a + p is invertible and a(2)p is quasinilpotent. The ring R is called strongly quasipolar in case each of its elements is strongly quasipolar. Some basic properties of the strongly quasipolar rings are obtained. The class of strongly quasipolar rings lies properly between the classes of pseudopolar rings and quasipolar rings. We determine the conditions under which a quasipolar ring is strongly quasipolar. We also show that strongly quasipolarity is a generalization of uniquely cleanness. When we consider this concept in terms of generalized inverses, we get that every pseudo Drazin invertible element is strongly quasipolar, and every strongly quasipolar element is generalized Drazin invertible.