Extensions of quasipolar rings

An associative ring with identity is called quasipolar provided that for each a is an element of R there exists an idempotent p is an element of R such that p is an element of comm(2)(a), a + p is an element of U(R) and ap is an element of R-qnil. In this article, we introduce the notion of quasipolar general rings (with or without identity). Some properties of quasipolar general rings are investigated. We prove that a general ring I is quasipolar if and only if every element a is an element of I can be written in the form a = s + q where s is strongly regular, s is an element of comm(2) (a), q is quasinilpotent, and sq = qs = 0. It is shown that every ideal of a quasipolar general ring is quasipolar. Particularly, we show that R is pseudopolar if and only if R is strongly pi-rad clean and quasipolar.