Abstract:In this paper we introduce the concept of $\tau$-extending modules by $\tau$-rational submodules and study some properties of such modules. It is shown that the set of all $\tau$-rational left ideals of $_RR$ is a Gabriel filter. An $R$-module $M$ is called $\tau$-extending if every submodule of $M$ is $\tau$-rational in a direct summand of~ $M$. It is proved that $M$ is $\tau$-extending if and only if $M = Rej_ME(R/\tau(R))\oplus N$, such that $N$ is a $\tau$-extending submodule of $M$. An example is given to show that the direct sum of $\tau$-extending modules need not be $\tau$-extending.
Keywords: torsion theory; $\tau$-rational submodules; $\tau$-closed submodules; $\tau$-extending modules
DOI: DOI 10.14712/1213-7243.2015.172
AMS Subject Classification: 16D10 16D80 16D99