A note on the dual of a finitely generated multiplication module
Let R be a commutative ring and M a finitely generated \((= f.g.)\) multiplication R-module, i.e., every submodule N of M has the form JM for some ideal J of R. Putting \(D=AnnAnn(M)\) and \(M^*=\) the dual of M, the authors show: Theorem. Let M be a f.g. multiplication module, and assume that D is a f.g. projective (or flat) ideal, then \(M^*\) is a f.g. projective (resp. flat) module. Corollary. With assumptions as above, if \(Ann(M)=Ann(D)\), then M is projective. [See also the following review Zbl 0685.13004.]
finitely generated multiplication module
scientific article; zbMATH DE number 4122093