Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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GRADED w-NOETHERIAN MODULES OVER GRADED RINGS

  • Wu, Xiaoying (School of Mathematics and Science Sichuan Normal University)
  • Received : 2019.11.10
  • Accepted : 2020.07.09
  • Published : 2020.09.30
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Abstract

In this paper, we study the basic theory of the category of graded w-Noetherian modules over a graded ring R. Some elementary concepts, such as w-envelope of graded modules, graded w-Noetherian rings and so on, are introduced. It is shown that: (1) A graded domain R is graded w-Noetherian if and only if Rg𝔪 is a graded Noetherian ring for any gr-maximal w-ideal m of R, and there are only finite numbers of gr-maximal w-ideals including a for any nonzero homogeneous element a. (2) Let R be a strongly graded ring. Then R is a graded w-Noetherian ring if and only if Re is a w-Noetherian ring. (3) Let R be a graded w-Noetherian domain and let a ∈ R be a homogeneous element. Suppose 𝖕 is a minimal graded prime ideal of (a). Then the graded height of the graded prime ideal 𝖕 is at most 1.

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Acknowledgement

The author would like to thank the referee for a careful reading and relevant comments. This work was partially supported by the National Natural Science Foundation of China (11671283, 11661014 and 11961050) and the Guangxi Natural Science Foundation (2016GXSFDA380017).

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