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.