Quasipolarity of Generalized Matrix Rings

TL;DR

This paper studies the quasipolarity property of generalized matrix rings over local rings, establishing conditions under which these rings are quasipolar and linking it to their structural properties.

Contribution

It characterizes quasipolarity in generalized matrix rings $K_s(R)$ over local rings, including conditions involving trace, determinant, and solvability of quadratic equations.

Findings

$K_s(R)$ is quasipolar if $s$ is nilpotent.

Quasipolarity of $K_s(R)$ depends on trace and quadratic equation solvability.

$M_2(R)$ is quasipolar iff it is strongly clean.

Abstract

An element $a$ of a ring $R$ is called \emph{quasipolar} provided that there exists an idempotent $p\in R$ such that $p\in comm^2(a)$, $a+p\in U(R)$ and $ap\in R^{qnil}$. 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 $tr(A)\in J(R)$ or the equation $x^2-tr(A)x+det_s(A)=0$ is solvable in $R$ for every $A\in K_s(R)$ with $det_s(A)\in J(R)$. Furthermore, we prove that $M_2(R)$ is quasipolar if and only if $M_2(R)$ is strongly clean for a commutative local ring $R$.