Gangyong Lee
Chungnam National University
Title
The relatively dense properties
Abstract
In 1958, G.D. Findlay and J. Lambek [1] defined a relationship among three modules over a ring $R$, $A_R\leq B_R (C_R)$, meaning that $A_R$ is a relatively dense submodule of $B_R$ to $C_R$, for studying the maximal right ring of quotients of a ring which is an extended ring of the base ring.
In this talk, we introduce the relatively dense properties. As an application of the relatively dense property, we show that the endomorphism ring of a module is embedded into the endomorphism ring of its rational hull. Also, 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 $M$ is called \emph{relatively dense to a right $R$-module} $K$ (or $K$-\emph{dense}) in $M$ if for any $m\in M$ and $0\neq k\in K$, $k\cdot m^{-1}N\neq 0$. In addition, we provide a condition such that $\text{End}_R(M)=\text{End}_H(M)$ where $H$ is a right ring of quotients of a ring $R$. This condition is that $R$ is $M_R$-dense in $H_R$.
[1] G.D. Findlay; J. Lambek, A generalized ring of quotients I, Canad. Math. Bull., 1958, 1, 77--85
[2] G. Lee, The rational hull of modules, Bulletin of the Malaysian Mathematical Sciences Society, 2024, 47(5), 164 (15 pages), DOI:10.1007/s40840-024-01759-4
[3] X. Zhang; G. Lee; N.K. Tung, Relatively polyform modules, J. Algebra Appl., 2024, 23(8), 2550046 (17 pages), DOI:10.1142/S021949882550046X
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