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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| Format: | Article |
| Language: | English |
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Wiley
2022-01-01
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| Series: | Journal of Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2022/7403502 |
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| author | De Chuan Zhou Hwankoo Kim Xiaolei Zhang Jin Xie |
| author_facet | De Chuan Zhou Hwankoo Kim Xiaolei Zhang Jin Xie |
| author_sort | De Chuan Zhou |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-871ee02c5b454317a8bef8d64688bc0c |
| institution | Kabale University |
| issn | 2314-4785 |
| language | English |
| publishDate | 2022-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Mathematics |
| spelling | doaj-art-871ee02c5b454317a8bef8d64688bc0c2025-02-03T05:58:00ZengWileyJournal of Mathematics2314-47852022-01-01202210.1155/2022/7403502Some Characterizations of w-Noetherian Rings and SM RingsDe Chuan Zhou0Hwankoo Kim1Xiaolei Zhang2Jin Xie3School of Mathematics and PhysicsDivision of Computer EngineeringSchool of Mathematics and SciencesSchool of Mathematics and PhysicsIn 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.http://dx.doi.org/10.1155/2022/7403502 |
| spellingShingle | De Chuan Zhou Hwankoo Kim Xiaolei Zhang Jin Xie Some Characterizations of w-Noetherian Rings and SM Rings Journal of Mathematics |
| title | Some Characterizations of w-Noetherian Rings and SM Rings |
| title_full | Some Characterizations of w-Noetherian Rings and SM Rings |
| title_fullStr | Some Characterizations of w-Noetherian Rings and SM Rings |
| title_full_unstemmed | Some Characterizations of w-Noetherian Rings and SM Rings |
| title_short | Some Characterizations of w-Noetherian Rings and SM Rings |
| title_sort | some characterizations of w noetherian rings and sm rings |
| url | http://dx.doi.org/10.1155/2022/7403502 |
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