Quasipolar Subrings of $3\times 3$ Matrix Rings

TL;DR

This paper characterizes when subrings of 3x3 matrix rings over local rings are quasipolar, linking properties of the base ring to the quasipolarity of matrix subrings, and extends results to power series rings.

Contribution

It establishes necessary and sufficient conditions for subrings of 3x3 matrices over local rings to be quasipolar, especially relating to bleached and uniquely bleached rings, and extends to power series rings.

Findings

$ ext{T}_3(R)$ is quasipolar iff $R$ is uniquely bleached.

Quasipolarity of $ ext{T}_n(R)$ is equivalent to that of $ ext{T}_n(R[[x]])$.

Characterizes quasipolar subrings over local rings.

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 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 $n$.