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http://dx.doi.org/10.14317/jami.2020.271

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)
Publication Information
Journal of applied mathematics & informatics / v.38, no.3_4, 2020 , pp. 271-276 More about this Journal
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.
Keywords
${\ast}$-Noetherian spectrum; ${\ast}$-finite ideal; radical operation; finite character;
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