Direct sum behavior of lattices over sigma-I rings (Q1109832)

A ring-order is a commutative Noetherian reduced ring R of Krull dimension one whose integral closure is a finitely generated R-module. A \(\Sigma\) I-ring is a ring-order in which every lattice (i.e. finitely generated submodule of a free R-module) is isomorphic to a direct sum of ideals. For example, a Bass ring (i.e. a ring-order in which every ideal is doubly generated) is a \(\Sigma\) I-ring. This paper considers what uniqueness is inherent in a lattice's direct sum decomposition into undecomposable lattices. It is shown that a \(\Sigma\) I-ring has the property that every lattice has a unique decomposition into a direct sum of indecomposable lattices if and only if R has at most one singular maximal ideal and Pic(R) is trivial. To handle the global case, the genus class group G(M) of a lattice M is studied. It is shown that G(M) is a homomorphic image of a direct sum of t copies of Pic(R) where t depends only on R. The \(\Sigma\) I-rings in which every lattice has a unique number of indecomposables summands are characterized.