컨텐츠 시작
학술대회/행사
초록검색
| 제출번호(No.) | 0571 |
|---|---|
| 분류(Section) | Poster Session |
| 분과(Session) | Algebra / Representation Theory / Lie Theory (SS-02) |
| 영문제목 (Title(Eng.)) |
The relationship of the closeness, denseness, and essentiallity of a submodule |
| 저자(Author(s)) |
Sri Wahyuni1 Dept. of Mathematics, Fact. of Mathematics and Natural Sciences, Universitas Gadjah Mada, Yogyakarta, Indonesia.1 |
| 초록본문(Abstract) | Let $R$ be a ring, $M$ be an $R$ module, and $S$ is a submodule of $M$. Submodule $S$ is called {\bf closed} if and only if $S=Cl(S)$, where $Cl(S):=\{m\in M \mid \exists r\in R-\{0\} \; {\rm such \; that} \; rm \in S\}$. $Cl(S)$ is called the {\bf closure} of $S$. It can be shown that $Cl(S)$ is the smallest closed submodule in $M$ containing $S$. For the submodule $S$, an element $y \in M$, define $\frac{1}{y}S:=\{r\in R \mid ry \in S\}$. It can be shown that $\frac{1}{y}S$ is an ideal in $R$ and $(\frac{1}{y}S) S \subseteq S$. Submodule $S$ is said to be {\bf dense} if for all $x$ and $y$ in M with $x\neq 0$, there exist an $r$ in $R$ such that $xr\neq 0$ and $ry$ is in $S$. In other words, using the introduced notation, the set $(\frac{1}{y}S) x\neq\{0\}$. In this case, the relationship is denoted by $S\subseteq_dM$. Further for submodule $S$, the module $M$ is said to be an {\bf essential extension} of $S$ is for every submodule $N$ of $M$, if $S \bigcap N=\{0\}$ then $N=\{0\}$. In this case, the relationship is denoted by $S\subseteq_eM$. It will be presented the relationship between the above concepts in term of the characterization of a submodule $S$ becoming a direct summand in $R$-module $M$. |
| 분류기호 (MSC number(s)) |
13A15, 13G05 |
| 키워드(Keyword(s)) | closed submodule, dense submodule, essential extention |
| 강연 형태 (Language of Session (Talk)) |
English |