Nil-quasipolar rings

Let R be an arbitrary ring. An element a is an element of R is nil-quasipolar if there exists p(2) = p is an element of comm(2)(a) such that a + p is an element of 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 2x2 matrix over a commutative local ring is nil-quasipolar. It is shown that A is an element of M-2(R) is nil-quasipolar if and only if A is an element of Nil(M-2(R)) or A + I-2 is an element of Nil (M-2(R)) or the characteristic polynomial chi(A) has a root in Nil(R) and a root in -1 + Nil(R). Wegive 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.