An approach to quasipolarity for rings along nilpotent elements

In this paper, we deal with a new approach to quasipolarity notion for rings, namely an element a of a ring R is called weakly nil-quasipolar if there exists \(p^2 = p\in comm^2(a)\) such that \(a + p\) or \(a-p\) is nilpotent, and the ring R is called weakly nil-quasipolar if every element of R is weakly nil-quasipolar. The class of weakly nil-quasipolar rings lies properly between the classes of nil-quasipolar rings and quasipolar rings. Although it is an open problem whether strongly clean (even quasipolar) rings have stable range one, we show that there is an affirmative answer for weakly nil-quasipolar rings. It is also proved that if R is a weakly nil-quasipolar NI ring, then R / N(R) is commutative. Moreover, we consider the question of when certain \(2 \times 2\) matrices over a commutative local ring is weakly nil-quasipolar.