Commutative semi-coherent and semi-regular rings

Abstract

Throughout this paper all rings considered will be commutative rings with identity. We shall consider the consequences for a ring of the flatness of certain classes of its modules. Since flatness is determined by the finitely generated ideals of a ring we shall in a real sense be studying conditions on these ideals. A coherent ring is a ring whose finitely generated ideals are finitely presented. Chase has proved [3, Theorem 2.11 that a ring is coherent if and only if any direct product of its flat modules is flat. In [8, Theorem 1 J we proved that a ring R is coherent if and only if Hom,(B, C) is flat whenever B and C are injective R-modules. We define a ring R to be semicoherent if Hom,(B, C) is a submodule of a flat R-module whenever B and C are injective R-modules. This is a property shared by both integral domains and Noetherian rings. We prove in Proposition 1.1 that a reduced ring R is semi-coherent if and only if min R is compact. Proposition 1.3 states that if R is a semi-coherent ring, then a finitely generated submodule of a free R-module is flat if and only if it is projective. A regular ring is a ring whose finitely generated ideals are direct summands of the ring. Equivalently it is a ring such that every one of its modules is flat. Inspired by our definition of semi-coherent we define a ring to be semi-regular if every module is a submodule of a flat module. Proposition 2.2 states that if the total ring of quotients of a ring R is semiregular, then R is a semi-coherent ring. An even stronger connection with the notion of coherence is provided by Proposition 2.3: A ring R is semiregular if and only if R is coherent and RM is semi-regular for every maximal ideal M of R. Proposition 2.7 states that if R is a reduced ring, then R is semi-regular if and only if R is regular; and Proposition 3.4 states that if R is a Noetherian