Quasipolarity of Generalized Matrix Rings

Orhan Gurgun; Sait Halicioglu; Abdullah Harmanci

paper

Quasipolarity of Generalized Matrix Rings

arXiv:1303.3173

Abstract

An element $a$ of a ring $R$ is called \emph{quasipolar} provided that there exists an idempotent $p \in R$ such that $p \in co m m^{2} (a)$ , $a + p \in U (R)$ and $a p \in R^{q ni l}$ . A ring $R$ is \emph{quasipolar} in case every element in $R$ is quasipolar. In this paper, we investigate quasipolarity of generalized matrix rings $K_{s} (R)$ for a commutative local ring $R$ and $s \in R$ . We show that if $s$ is nilpotent, then $K_{s} (R)$ is quasipolar. We determine the conditions under which elements of $K_{s} (R)$ are quasipolar. It is shown that $K_{s} (R)$ is quasipolar if and only if $t r (A) \in J (R)$ or the equation $x^{2} - t r (A) x + d e t_{s} (A) = 0$ is solvable in $R$ for every $A \in K_{s} (R)$ with $d e t_{s} (A) \in J (R)$ . Furthermore, we prove that $M_{2} (R)$ is quasipolar if and only if $M_{2} (R)$ is strongly clean for a commutative local ring $R$ .

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