Bulletin of the
Korean Mathematical Society BKMS

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