Nil-quasipolar rings

Gurgun O., HALICIOĞLU S., Harmanci A.

BOLETIN DE LA SOCIEDAD MATEMATICA MEXICANA, vol.20, no.1, pp.29-38, 2014 (ESCI) identifier

  • Publication Type: Article / Article
  • Volume: 20 Issue: 1
  • Publication Date: 2014
  • Doi Number: 10.1007/s40590-014-0005-y
  • Journal Name: BOLETIN DE LA SOCIEDAD MATEMATICA MEXICANA
  • Journal Indexes: Emerging Sources Citation Index (ESCI)
  • Page Numbers: pp.29-38
  • Keywords: Nil-quasipolar matrix, Quasipolar ring, Strongly nil-clean ring, Matrix ring, Characteristic polynomial
  • Ankara University Affiliated: Yes

Abstract

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.