Modules (Algebra)

Birkhoff raised the question of how to determine "relative invariants of subgroups" of a given group. Let us consider pairs (A, B ) where B is a finite pn-bounded Abelian group and A is a subgroup of B. Maps between pairs (A, B) --> (A', B') are morphisms f : B --> B' such that f (A) --> A'. Classification of such pairs, up to isomorphism, is Birkhoff's famous problem. By the Krull-Remak-Schmidt theorem, an arbitrary pair (A, B) is a direct sum of indecomposable pairs, and the multiplicities of the indecomposables are determined uniquely. The purpose of this thesis is to describe the decomposition of such pairs, (A, B), explicitly for n = 2 and n = 3. We describe explicitly how an indecomposable pair can possibly embed into a given pair (A, B). This construction gives rise to formulas for the multiplicity of an indecomposable in the direct sum decomposition of the pair (A, B). These decomposition numbers form a full set of relative invariant, as requested by Birkhoff.

We say that a commutative ring R has the unique decomposition into ideals (UDI) property if, for any R-module which decomposes into a finite direct sum of indecomposable ideals, this decomposition is unique up to the order and isomorphism class of the ideals. In a 2001 paper, Goeters and Olberding characterize the UDI property for Noetherian integral domains. In Chapters 1-3 the UDI property for reduced Noetherian rings is characterized. In Chapter 4 it is shown that overrings of one-dimensional reduced commutative Noetherian rings with the UDI property have the UDI property, also. In Chapter 5 we show that the UDI property implies the Krull-Schmidt property for direct sums of torsion-free rank one modules for a reduced local commutative Noetherian one-dimensional ring R.

The structure of minimal zero-dimensional extension of rings with Noetherian spectrum in which zero is a primary ideal and with at most one prime ideal of height greater than one is determined. These rings include K[[X,T]] where K is a field and Dedenkind domains, but need not be Noetherian nor integrally closed. We show that for such a ring R there is a one-to-one correspondence between isomorphism classes of minimal zero-dimensional extensions of R and sets M, where the elements of M are ideals of R primary for distinct prime ideals of height greater than zero. A subsidiary result is the classification of minimal zero-dimensional extensions of general ZPI-rings.