Finiteness in prime ideals in rings of global dimension two

Abstract

Let A A be a commutative ring with identity. The main result states conditions that ensure the finiteness of prime ideals in a coherent ring A A of global dimension two. Precisely, any ideal containing two noncomparable prime ideals is finitely generated. As a corollary it follows that a Krull domain of global dimension two is noetherian. Another corollary is that if A A is not semihereditary it contains a finitely generated maximal ideal.