Monoids and direct-sum decompositions over local rings

✍ Scribed by K. Kattchee

• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 148 KB
• Volume: 256
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1016/s0021-8693(02)00030-3

✦ Synopsis

Let R be a local ring order, i.e. a one-dimensional local (noetherian) ring whose completion R 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 (b 1 , . . . , b n ) such that the R-module n i=1 b i V i is extended from an R-module, where V 1 , . . . , V n are the distinct (and uniquely determined) indecomposable direct summands of the R-module M. Here bV denotes the direct sum of b copies of V , and N = {0, 1, 2, . . .}. The monoids +(M) and Λ(M) are isomorphic, and we show that Λ(M) = ker(A) ∩ N n for some integer matrix A ∈ Z d×n . Monoids which are isomorphic to ker(A) ∩ N n for some A ∈ Z d×n 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 Γ ∼ = ker(A) ∩ N n , where A ∈ Z d×n .