Quasipolarity of Generalized Matrix Rings

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 $R$.