Some Characterizations of w-Noetherian Rings and SM Rings

In this paper, we characterize w-Noetherian rings and SM rings. More precisely, in terms of the u-operation on a commutative ring R, we prove that R is w-Noetherian if and only if the direct limit of rGV-torsion-free injective R-modules is injective and that R is SM, which can be regarded as a regul...

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Bibliographic Details
Published in:Journal of mathematics (Hidawi) 2022-01, Vol.2022 (1)
Main Authors: Zhou, De Chuan, Kim, Hwankoo, Zhang, Xiaolei, Xie, Jin
Format: Article
Language:English
Subjects:
Mathematics
Modules
Rings (mathematics)
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Summary:In this paper, we characterize w-Noetherian rings and SM rings. More precisely, in terms of the u-operation on a commutative ring R, we prove that R is w-Noetherian if and only if the direct limit of rGV-torsion-free injective R-modules is injective and that R is SM, which can be regarded as a regular w-Noetherian ring, if and only if the direct limit of GV-torsion-free (or rGV-torsion-free) reg-injective R-modules is reg-injective. As a by-product of the proof of the second statement, we also obtain that the direct and inverse limits of u-modules are both u-modules and that SM rings are regular w-coherent.
ISSN:2314-4629
2314-4785
DOI:10.1155/2022/7403502