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Volume 31, Issue 3
Pseudopolarity of Generalized Matrix Rings over a Local Ring

Xiaobin Yin & Dou Wan

DOI: 10.13447/j.1674-5647.2015.03.03

Commun. Math. Res., 31 (2015), pp. 211-221.

Published online: 2021-05

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  • Abstract

Pseudopolar rings are closely related to strongly $π$-regular rings, uniquely strongly 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 conditions under which elements of $K_s(R)$ are pseudopolar. Assume that $R$ is a local ring. It is shown 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 injective and $u ∈ U(R)$ and $j ∈ J(R)$. Furthermore, several equivalent conditions for $K_s(R)$ over a local ring $R$ to be pseudopolar are obtained.

  • Keywords

pseudopolar ring, local ring, generalized matrix ring.

  • AMS Subject Headings

16E50

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CMR-31-211, author = {Yin , Xiaobin and Wan , Dou}, title = {Pseudopolarity of Generalized Matrix Rings over a Local Ring}, journal = {Communications in Mathematical Research }, year = {2021}, volume = {31}, number = {3}, pages = {211--221}, abstract = {

Pseudopolar rings are closely related to strongly $π$-regular rings, uniquely strongly 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 conditions under which elements of $K_s(R)$ are pseudopolar. Assume that $R$ is a local ring. It is shown 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 injective and $u ∈ U(R)$ and $j ∈ J(R)$. Furthermore, several equivalent conditions for $K_s(R)$ over a local ring $R$ to be pseudopolar are obtained.

}, issn = {2707-8523}, doi = {https://doi.org/10.13447/j.1674-5647.2015.03.03}, url = {http://global-sci.org/intro/article_detail/cmr/18922.html} }
TY - JOUR T1 - Pseudopolarity of Generalized Matrix Rings over a Local Ring AU - Yin , Xiaobin AU - Wan , Dou JO - Communications in Mathematical Research VL - 3 SP - 211 EP - 221 PY - 2021 DA - 2021/05 SN - 31 DO - http://doi.org/10.13447/j.1674-5647.2015.03.03 UR - https://global-sci.org/intro/article_detail/cmr/18922.html KW - pseudopolar ring, local ring, generalized matrix ring. AB -

Pseudopolar rings are closely related to strongly $π$-regular rings, uniquely strongly 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 conditions under which elements of $K_s(R)$ are pseudopolar. Assume that $R$ is a local ring. It is shown 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 injective and $u ∈ U(R)$ and $j ∈ J(R)$. Furthermore, several equivalent conditions for $K_s(R)$ over a local ring $R$ to be pseudopolar are obtained.

Yin , Xiaobin and Wan , Dou. (2021). Pseudopolarity of Generalized Matrix Rings over a Local Ring. Communications in Mathematical Research . 31 (3). 211-221. doi:10.13447/j.1674-5647.2015.03.03
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