Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
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HILBERT BASIS THEOREM FOR RINGS WITH ∗-NOETHERIAN SPECTRUM

  • PARK, MIN JI (Department of Mathematics, College of Life Science and Nano Technology, Hannam University) ;
  • LIM, JUNG WOOK (Department of Mathematics, College of Natural Sciences, Kyungpook National University)
  • Received : 2019.12.17
  • Accepted : 2020.01.20
  • Published : 2020.05.30
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Abstract

Let R be a commutative ring with identity, R[X] the polynomial ring over R, ∗ a radical operation on R and ⋆ a radical operation of finite character on R[X]. In this paper, we give Hilbert basis theorem for rings with ∗-Noetherian spectrum. More precisely, we show that if (I∗R[X])⋆ = (IR[X])⋆ and (I∗R[X])⋆ ∩ R = I∗ for all ideals I of R, then R has ∗-Noetherian spectrum if and only if R[X] has ⋆-Noetherian spectrum. This is a generalization of a well-known fact that R has Noetherian spectrum if and only if R[X] has Noetherian spectrum.

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References

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