컨텐츠 시작
학술대회/행사
초록검색
| 제출번호(No.) | 0240 |
|---|---|
| 분류(Section) | Special Session |
| 분과(Session) | (SS-03) Commutative Algebra and Related Topics (SS-03) |
| 발표시간(Time) | 20th-D-13:00 -- 13:20 |
| 영문제목 (Title(Eng.)) |
On relatively dense modules |
| 저자(Author(s)) |
Gangyong Lee1 Chungnam National University1 |
| 초록본문(Abstract) | The theory of rings of quotients has its origin in the work of \O. Ore \cite{oo} and K. Asano \cite{as} on the construction of the total ring of fractions, in the 1930's and 40's. But the subject did not really develop until the end of the 1950's, when a number of important papers appeared (by R.E. Johnson, Y. Utumi, A.W. Goldie, J. Lambek et al). In particular, Johnson(1951), Utumi(1956), and Findlay $\&$ Lambek(1958) have studied the maximal right ring of quotients of a ring which is an extended ring of the base ring. In 1958, G.D. Findlay and J. Lambek \cite{fl} defined a relationship among three modules over a ring $R$, $A_R\leq B_R (C_R)$, which means that $A_R$ is a relatively dense submodule of $B_R$ to $C_R$. In this talk, we introduced the relatively dense property and the relatively polyform property. As an application of the relatively dense property, the equivalent condition for the rational hull of the finite direct sum of modules to be the direct sum of the rational hulls of those modules is shown. Note that a submodule $N$ of a right $R$-module $N$ is called \emph{relatively dense to a right $R$-module} $K$ (or $K$-\emph{dense}) in $M$ if for any $m\in M$ and $0\neq k\in K$, $k\cdot m^{-1}N\neq 0$. Also, we provide a condition to be $\text{End}_R(M)=\text{End}_H(M)$ where $H$ is a right ring of quotients of a ring $R$. This condition is that $R$ is $M_R$-dense in $H_R$. Additionally, we show that a right $R$-module $M$ is nonsingular if and only if $R$ is $M$-polyform. Note that $R$ is called \emph{$M$-polyform} if every essential ideal of $R$ is $M$-\emph{dense}. \begin{thebibliography}{60} \bibitem{as} K. Asano, \"{U}ber die Quotientenbildung von Schiefringen, J. Math. Soc. Japan, {\bf 1949} \emph{1(2)}, 73--78 \bibitem{fl} G.D. Findlay; J. Lambek, A generalized ring of quotients I, Canad. Math. Bull., {\bf 1958} \emph{1}, 77--85 \bibitem{lee2} G. Lee, The rational hull of modules, Submitted \bibitem{oo} \O. Ore, Linear equations in non-commutative fields, Ann. of Math. Second Series, {\bf 1931} \emph{32(3)}, 463--477 \end{thebibliography} |
| 분류기호 (MSC number(s)) |
16D70, 16S50 |
| 키워드(Keyword(s)) | Maximal right ring of quotients, dense module, polyform module, relatively dense module, relatively polyform module |
| 강연 형태 (Language of Session (Talk)) |
Korean |