Big Indecomposable Modules And Direct-Sum Relations

A commutative Noetherian local ring (R, m) is said to be Dedekind-like provided R has Krull-dimension one, R has no non-zero nilpotent elements, the integral closure (R) over bar of R is generated by two elements as an R-module, and m is the Jacobson radical of (R) over bar. A classification theorem due to Klingler and Levy implies that if M is a finitely generated indecomposable module over a Dedekind-like ring, then, for each minimal prime ideal P of R, the vector space M-P has dimension 0, 1 or 2 over the field R-P. The main theorem in the present paper states that if R (commutative, Noetherian and local) has non-zero Krull dimension and is not a homomorphic image of a Dedekind-like ring, then there are indecomposable modules that are free of any prescribed rank at each minimal prime ideal.