A Krull-Schmidt theorem for Noetherian modules
Direct sum cancellation is known to fail for nonzero Noetherian modules in general: if \(A,B,C\) satisfy \(A\oplus C\cong B\oplus C\), then \(A\) is not necessarily isomorphic to \(B\), unless \(A\), \(B\), and \(C\) have finite length. In the latter case, this follows from the classical Krull-Schmidt theorem which asserts that a finite length module can be written uniquely (up to order of summands) as a direct sum of indecomposable modules. However, the Krull-Schmidt theorem does not apply to the entire category of Noetherian modules because endomorphism rings of indecomposable modules are not always local. The author circumvents this difficulty by introducing a new category whose objects are Noetherian modules, but whose morphisms are defined in such a way that an adaptation of Fitting's lemma can be used to show that indecomposable objects in this category do indeed have local endomorphism rings. The role of the length of a module in the classical version of Fitting's lemma is here played by the ordinal length of a Noetherian module as defined by \textit{T.~H.~Gulliksen} [J. Pure Appl. Algebra 3, 159-170 (1973; Zbl 0271.13005)], and the main theorem can be stated as follows. If \(A,B,C\) are Noetherian modules such that either \(A\oplus C\cong B\oplus C\) or \(A^n\cong B^n\) for some natural number \(n\), then there exist submodules \(A'\subseteq A\), \(B'\subseteq B\) whose (ordinal) lengths are equal to those of \(A\) and \(B\), respectively, such that \(A'\cong B'\).
scientific article; zbMATH DE number 1797463
scientific article