An Anderson's theorem on noncommutative rings
Bull. Korean Math. Soc. 2008 Vol. 45, No. 4, 797-800
Printed December 1, 2008
Chan Huh, Nam Kyun Kim, and Yang Lee
Busan National University, Hanbat National University, and Busan National University
Abstract : Let $R$ be a ring and $I$ be a proper ideal of $R$. For the case of $R$ being commutative, Anderson proved that $(*)$ there are only finitely many prime ideals minimal over $I$ whenever every prime ideal minimal over $I$ is finitely generated. We in this note extend the class of rings that satisfies the condition $(*)$ to noncommutative rings, so called homomorphically IFP, which is a generalization of commutative rings. As a corollary we obtain that there are only finitely many minimal prime ideals in the polynomial ring over $R$ when every minimal prime ideal of a homomorphically IFP ring $R$ is finitely generated.
Keywords : commutative ring, (homomorphically) IFP ring, minimal prime ideal
MSC numbers : 16D25, 16N60
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