Abstract :
Let R be a local ring order, i.e. a one-dimensional local (noetherian) ring whose completion is reduced, and let M be a finitely generated R-module. We consider two monoids: +(M), which consists of the isomorphism classes of R-modules which arise as direct summands of direct sums of finitely many copies of M, and Λ(M), which consists of the n-tuples (b1,…,bn) such that the -module i=1nbiVi is extended from an R-module, where V1,…,Vn are the distinct (and uniquely determined) indecomposable direct summands of the -module . Here bV denotes the direct sum of b copies of V, and . The monoids +(M) and Λ(M) are isomorphic, and we show that for some integer matrix . Monoids which are isomorphic to for some are called positive normal. In [R. Wiegand, J. Algebra, in press] it is shown that given a positive normal monoid Γ, there exist a local ring-order domain R and finitely generated torsion-free R-module M such that Γ Λ(M). We show that given a local ring order R, there exists a positive normal monoid Γ such that for each finitely generated R-module M, Λ(M) is not isomorphic to Γ. The proof depends on the fact that there exist rank-three positive normal monoids with arbitrarily large embedding dimension, where the embedding dimension of Γ is defined as the smallest n such that , where .
