Quasipolar Subrings of $3\times 3$ 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 determine conditions under which subrings of $3\times 3$ matrix rings over local rings are quasipolar. Namely, if $R$ is a bleached local ring, then we prove that $\mathcal{T}_3(R)$ is quasipolar if and only if $R$ is uniquely bleached. Furthermore, it is shown that $T_n(R)$ is quasipolar if and only if $T_n\big(R[[x]]\big)$ is quasipolar for any positive integer $n$.