Unique decomposition into ideals for commutative Noetherian rings

Abstract

A commutative ring R has the unique decomposition into ideals (UDI) property if, for any R-module that decomposes into a finite direct sum of indecomposable ideals, this decomposition is unique up to the order and isomorphism classes of the indecomposable ideals. In a 2001 paper, Goeters and Olberding characterize the UDI property for Noetherian integral domains, noting the almost local nature of the property and, in the case of a local domain, relating it to the structure of its integral closure. In a 2011 paper, Ay and Klingler obtain similar results for Noetherian reduced rings. In this paper, we examine the UDI property for arbitrary commutative Noetherian rings, establishing the same almost local nature of the property, and giving an example which shows that the local results do not extend to commutative Noetherian rings in general.