Bulletin of the
Korean Mathematical Society BKMS

In this paper, we introduce and investigate a new class of modules that is closely related to the class of Noetherian modules. Let $R$ be a commutative ring and $M$ be an $R$-module. We say that $M$ is an $r$-Noetherian module if every $r$-submodule of $M$ is finitely generated. Also, we call the ring $R$ to be an $r$-Noetherian ring if $R$ is an $r$-Noetherian $R$-module, or equivalently, every $r$-ideal of $R$ is finitely generated. We show that many properties of Noetherian modules are also true for $r$-Noetherian modules. Moreover, we extend the concept of weakly Noetherian rings to the category of modules and we characterize Noetherian modules in terms of $r$-Noetherian and weakly Noetherian modules. Finally, we use the idealization construction to give non-trivial examples of $r$-Noetherian rings that are not Noetherian.

Keywords: $r$-Noetherian module, $r$-Noetherian ring, $r$-submodule, $r$-ideal, weakly Noetherian module, weakly Noetherian ring, Noetherian module, Noetherian ring, idealization

MSC numbers: Primary 13E05; Secondary 13A15, 13G05, 13B30