On finitely generated projective modules and multiplication modules

Abstract

L e t R be a commuta t ive ring wi th 1. I f I is a f ini tely generated (f.g.) ideal in R, then I is a projective ideal iff I is a mult ipl icat ion ideal and ann (I) is generated by an idempotent [9]. The main goal of this paper is to s t u d y the relationship between f.g. projective modules and mult ipl icat ion modules. Recall t h a t a r ight module P is said to be a multiplication module if every submodule N of P has the form P I for some ideal I of R [1]. On the other hand, a f.g. projective module P is called heredi tar i ly projective if every homomorphic image of P into a f.g. projective module is projective. Equivalent ly, if l(P) is a projective ideal of R for each linear functional I on P (i.e. l E P*, the dual of P). The projective module P is called cohereditarily projective if the closure of each f.g. submodule N of P is a direct summand in P [2]. The main result of this paper is tha t for indecomposable f.g. modules P over P.P. rings R, P is heredi tar i ly projective iff P is a mult ipl icat ion module and ann (P) is generated by an idempotent (see Theorem 2.7). I t was proved in [2] t h a t a f.g. module P is heredi tar i ly projective iff P* is cohereditar i ly projective. I t is proved in this paper t h a t these two concepts are equivalent if P is indecomposable and R is a P.P. ring (see Theorem 2.8). Final ly , we remark t h a t all rings in this paper are commuta t ive with 1, and all modules are uni ta ry . For basic ideas on mult ipl icat ion ideals see [4].