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∈N with AnnM(a)=(0) implies that m∈N for each a∈R and m∈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 AnnM(a)≠(0) or aM=M, for every a∈R) and strongly uz-module (i.e., an R-module M such that aM⊆a2M, for every a∈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.