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Some Characterizations of w-Noetherian Rings and SM Rings

Author

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  • De Chuan Zhou
  • Hwankoo Kim
  • Xiaolei Zhang
  • Jin Xie
  • Marco Fontana

Abstract

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.

Suggested Citation

  • De Chuan Zhou & Hwankoo Kim & Xiaolei Zhang & Jin Xie & Marco Fontana, 2022. "Some Characterizations of w-Noetherian Rings and SM Rings," Journal of Mathematics, Hindawi, vol. 2022, pages 1-11, December.
    • Handle: RePEc:hin:jjmath:7403502
    DOI: 10.1155/2022/7403502
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