r-Submodules and uz-modules

In this article we study and investigate the behavior of r-submodules (a proper submodule N of an R-module M  in which am\in N with {\rm Ann}_M(a)=(۰) implies that m\in N for each a\in R and m\in M). We show that every simple submodule, direct summand,  divisible submodule, torsion submodule and the socle of a module is an r-submodule and if R is a domain, then the singular submodule is an r-submodule. We also introduce the concepts of uz-module (i.e., an R-module M such that either {\rm Ann}_M(a)\not=(۰) or aM=M, for every a\in R) and strongly  uz-module (i.e., an R-module M such that aM\subseteq a^۲M, for every a\in R) in the category of modules over commutative rings. We show that every Von Neumann regular module is a strongly uz-module and every Artinian  R-module is a   uz-module. It is observed that if M is a faithful cyclic R-module, then  M is a   uz-module if and only if every its cyclic submodule is an r-submodule. In addition, in this case, R is a domain if and only if the only r-submodule of M is zero submodule. Finally, we prove that R  is a uz-ring if and only if every faithful cyclic R-module is a uz-module.

کلیدواژه ها:

r-submodule ، uz-module ، strongly uz-module ، r-ideal