Pseudopolarity of Generalized Matrix Rings over a Local Ring (Communications in Mathematical Research, 2021)
Pseudopolar rings are closely related to strongly $π$-regular rings, uniquelystrongly clean rings and semiregular rings. In this paper, we investigate pseudopolarity of generalized matrix rings $K_s(R)$ over a local ring $R$. We determine the conditionsunder which elements of $K_s(R)$ are pseudopolar. Assume that $R$ is a local ring. It isshown that $\boldsymbol{A}∈ K_s(R)$ is pseudopolar if and only if $\boldsymbol{A}$ is invertible or $\boldsymbol{A}^2 ∈ J(K_s(R))$ or $\boldsymbol{A}$ is similar to a diagonal matrix $\begin{bmatrix} u & 0 \\ 0 & j \end{bmatrix}$, where $l_u −r_j$ and $l_j −r_u$ are injectiveand $u ∈ U(R)$ and $j ∈ J(R)$. Furthermore, several equivalent conditions for $K_s(R)$ over a local ring $R$ to be pseudopolar are obtained.