NEW ZEALAND JOURNAL OF MATHEMATICS Volume 34 (2005), 11–23 LOCALLY CYCLIC PROJECTIVE MODULES Majid M. Ali and David J. Smith (Received October 2003) Abstract. Let R be a commutative ring with identity and M an R–module. If M is either locally cyclic projective or faithful multiplication then M is locally either zero or isomorphic to R. We investigate locally cyclic projective modules and the properties they have in common with faithful multiplication modules. Our main tool is the trace ideal. We see that the module structure of a locally cyclic projective module and its trace ideal are closely related. We prove cancellation laws involving projective modules and their trace ideals. Among various applications, we show that the product of a prime ideal and a locally cyclic projective module is a prime submodule. Introduction All rings are commutative with identity and all modules are unital. Let R be a ring. An R–module M is a multiplication module if every submodule N of M has the form IM for some ideal I of R. Equivalently, N =[N : M ]M . Anderson [7] defined θ(M )= ∑ m∈M [Rm : M ] and showed that if M is multipli- cation then M = θ(M )M . Anderson and Al–Shaniafi also proved [5, Theorem 2.3] that if M is faithful multiplication then θ(M ) is a pure ideal of R ( equivalently θ(M ) is an idempotent multiplication ideal of R,[1] ) . Let M be a multiplica- tion module. Then M = θ(M )M . If m ∈ M then Rm =[Rm : M ]M =[Rm : M ]θ(M )M = θ(M )[Rm : M ]M = θ(M )m. The converse is also true: Suppose P is a maximal ideal of R and Rm = θ(M )m for each m ∈ M . If θ(M ) is con- tained in P then Rm = Pm, and hence (Rm) P =(Pm) P . By Nakayama’s Lemma (Rm) P =0 P , and hence M P =0 P . Otherwise θ(M )  P , and hence there exist m ∈ M and p ∈ P such that (1 − p)M ⊆ Rm. It follows by [6, Theorem 2.1] that M is multiplication. See also [3, Corollary 1.4(2)] and [20, Theorem 2]. The trace ideal of an R-module M is Tr(M )= ∑ f ∈Hom(M,R) f (M ). If M is projective then M = Tr(M )M , ann M = ann Tr(M ), and Tr(M ) is a pure ideal of R,[11, Proposition 3.30],[21], and [22]. Locally cyclic projective modules (for example projective ideals) and faithful multiplication modules have many common properties. If M is a locally cyclic projective module or a faithful multiplication module then M is locally either zero or isomorphic to R. If M is a locally cyclic projective (resp. faithful multiplication) R–module, then M is finitely generated if and only if Tr(M ) (resp. θ(M )) is finitely generated, [2, Corollary 2.6] and [5, Corollary 2.2]. More generally, if M is locally cyclic projective (resp. faithful multiplication), then for each r ∈ Tr(M ), 1991 Mathematics Subject Classification 13C13, 13A15. Key words and phrases: projective module, multiplication module, pure submodule, trace of a module, ring of endomorphisms.