Commutative rings and modules that are $r$-Noetherian
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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 nitely 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 nitely 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.
Abstract
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 nitely 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 nitely 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.
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