On the structure of maximal non-finitely generated ideals of ring and Cohen's theorem

In this paper we consider analogues of Cohen's theorem. We introduce new notions of almost prime left (right) submodule and $dr$-prime left (right) ideal, this allows us to extend Cohen's theorem for modular and non-commutative analogues. We prove that if every almost prime submodule of a finitely generated module is a finitely generated submodule, then any submodule of this module is finitely generated.