A generalization of primary ideals and strongly prime submodules

We present ∗-primary submodules, a generalization of the concept of primary submodules of an R-module. We show that every primary submodule of a Noetherian R-module is ∗-primary. Among other things, we show that over a commutative domain R, every torsion free R-module is ∗-primary. Furthermore, we show that in a cyclic R-module, primary and ∗-primary coincide. Moreover, we give a characterization of ∗-primary submodules for some finitely generated free R-modules.