Direct-sum decompositions of modules with semilocal endomorphism ring

Suppose C is a class of right R-modules with the following three properties: C has a set of representatives V(C) (that is, V(C) is a set, not a proper class); C is closed under finite direct sums, under direct summands and under isomorphisms; and End_R(A) is semilocal for each A in C. The first author showed that in this case V(C) is a reduced Krull monoid. Moreover, if M_R is the direct sum of the modules in V(C), E is the endomorphism ring End(M_R), S_E denotes the full subcategory of Mod-E consisting of finitely generated projective right E-modules with semilocal endomorphism ring, and C is viewed as a full subcategory of Mod-R, then the categories C and S_E turn out to be equivalent via the functors Hom_R(M_R,-): C -> S_E and -\otimes_E M: S_E -> C. In particular, the monoids V(C) and V(S_E) are isomorphic. The main theorem of this paper states that every reduced Krull monoid arises in this fashion, as V(S_R) for a suitable ring R. Thus reduced Krull monoids coincide both with the monoids that can be realized as V(C) for some class C of modules satisfying the three properties above, and with the monoids that can be realized as V(S_R) for some ring R.