Bulletin of the Korean Mathematical Society (대한수학회보)

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A GENERALIZATION OF ω-LINKED EXTENSIONS

  • Wu, Xiaoying (School of Mathematics Science Sichuan Normal University)
  • Received : 2021.06.01
  • Accepted : 2021.09.02
  • Published : 2022.05.31
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Abstract

In this paper, the concepts of ω-linked homomorphisms, the ω𝜙-operation, and DW𝜙 rings are introduced. Also the relationships between ω𝜙-ideals and ω-ideals over a ω-linked homomorphism 𝜙 : R → T are discussed. More precisely, it is shown that every ω𝜙-ideal of T is a ω-ideal of T. Besides, it is shown that if T is not a DW𝜙 ring, then T must have an infinite number of maximal ω𝜙-ideals. Finally we give an application of Cohen's Theorem over ω-factor rings, namely it is shown that an integral domain R is an SM-domain with ω-dim(R) ≤ 1, if and only if for any nonzero ω-ideal I of R, (R/I)ω is an Artinian ring, if and only if for any nonzero element α ∈ R, (R/(a))ω is an Artinian ring, if and only if for any nonzero element α ∈ R, R satisfies the descending chain condition on ω-ideals of R containing a.

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References

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