ON KRULL OVERRINGS OF A MAROT RING WHOSE REGULAR IDEALS ARE FINITELY GENERATED

Abstract

Let R be a Marot ring whose regular ideals are finitely generated and D a Krull overring of R. In this paper, we show that if reg-dimR less than or equal to 2, then each regular ideal of D is finitely generated and reg-dimD less than or equal to 2. In particular, each regular ideal of a Krull overring of a Noetherian ring R is finitely generated provided that (regular) Krull-dimension R less than or equal to 2. This is a generalization of the well-known fact that a Krull overring of a Noetherian domain with Krull-dimension less than or equal to 2 is also a Noetherian domain with Krull-dimension less than or equal to 2.