Nil-quasipolar rings

Abstract

Let \(R\) be an arbitrary ring. An element \(a\in R\) is nil-quasipolar if there exists \(p^2=p\in comm^2(a)\) such that \(a+p\in Nil(R)\); \(R\) is called nil-quasipolar in case each of its elements is nil-quasipolar. In this paper, we study nil-quasipolar rings over commutative local rings. We determine the conditions under which a single \(2\times 2\) matrix over a commutative local ring is nil-quasipolar. It is shown that \(A\in M_2(R)\) is nil-quasipolar if and only if \(A\in Nil\big (M_2(R)\big )\) or \(A+I_2\in Nil\big (M_2(R)\big )\) or the characteristic polynomial \(\chi (A)\) has a root in \(Nil(R)\) and a root in \(-1+Nil(R)\). We give some equivalent characterizations of nil-quasipolar rings through the endomorphism ring of a module. Among others we prove that every nil-quasipolar ring has stable range one.