컨텐츠 시작

학술대회/행사

초록검색

제출번호(No.) 0207
분류(Section) Special Session
분과(Session) (SS-04) Commutative Algebra and Related Topics (SS-04)
발표시간(Time) 25th-B-12:00 -- 12:20
영문제목
(Title(Eng.))
On rudimentary structural matrix rings
저자(Author(s))
Sangmin Chun2, Gangyong Lee1, Mauricio Medina-B\'{a}rcenas3, Nguyen Khanh Tung4
Chungnam National University1, Chung-Ang University2, Benemérita Universidad Autónoma de Puebla3, Vietnam National University4
초록본문(Abstract) In 2014, Lee, Roman, and Zhang defined rudimentary rings. We called a \emph{rudimentary ring} $R$ if there exists a faithful indecomposable endoregular module over the ring $R$.
At that time, they treated a partial matrix ring as a subring of $\mathsf{Mat}_n(\mathfrak{a})$
to give an example of rudimentary rings.
In 1988, Wyk \cite{wyk} defined the structural matrix ring as a subring of a full $n\times n$ matrix ring over a ring using a Boolean matrix and a binary relation (preorder).

In this talk, we provide that the structural matrix ring and the partial matrix ring coincide. We call a subset $\mathcal{A}$ of a full $n\times n$ matrix ring over $D$ an $n\times n$ \emph{structural matrix ring over $D$} if $\mathfrak{a}=\sum_{(i, j)\in\mathcal{R}}e_{ij}D$ where $e_{ij}$ are matrix units, denoted by $\mathsf{PM}_n(D)$.
where $D$ is a ring with unity and $\mathcal{R}$ is a preorder on the index set $\mathcal{I}=\{1, 2, \dots, n\}$.
Also, when $S$ is a subset of $MU(n)=\{e_{ij}\,|\,i, j\in \mathcal{I}\}$ containing $e_{11}, e_{22}, \dots, e_{nn}$,
we show that $S\cup\{0\}$ is a semigroup if and only if $\mathcal{R}=\{(i, j)\,|\,e_{ij}\in S\}$ is a preorder on $\mathcal{I}$. In addition, we provide that there is a bijective map from the set of all $n \times n$ structural matrix rings over a ring $D$ to the set of all transitive directed graphs $G=(V,E)$ with $V=\{1,\dots,n\}$.

\vspace{0.5cm}

This talk is based on a joint work with Sangmin Chun, Mauricio Medina-B\'arcenas, and Khanh Tung Nguyen.



\begin{thebibliography}{10}
\bibitem{lrz} G. Lee; C.S. Roman; X. Zhang, Modules whose endomorphism rings are division rings, Comm. Algebra, {\bf 2014} \emph{42(12)}, 5205--5223
\bibitem{wyk} L. van Wyk, Maximal left ideals in structural matrix rings, Comm. Algebra, {\bf 1988} \emph{16(2)}, 399--419
\end{thebibliography}
분류기호
(MSC number(s))
05C25, 15B99, 06A06
키워드(Keyword(s)) Structural matrix ring, transitive directed graph, preorder, partial order
강연 형태
(Language of Session (Talk))
Korean