On some weak versions of graded-Noetherian modules and rings

In this paper, we introduce and investigate two weak versions of graded-Noetherian modules. Let G be an abelian group with an identity element denoted by 0, R be a commutative G-graded ring and M be a G-graded R-module. We say that M is a graded-r-Noetherian module if every graded-r-submodule of M is finitely generated. Also, we say that M is a graded-weakly-Noetherian module if all finitely generated graded submodules of M are graded-Noetherian R-modules. We give many properties of the two different concepts and we examine the relation between them and the different concepts that already exist in the literature. We illustrate our study by giving many nontrivial examples and counter-examples. Moreover, we characterize graded-Noetherian modules in terms of graded-r-Noetherian and graded-weakly-Noetherian modules. Finally, we examine the transfer of these two concepts in the graded idealization of graded modules.