# Decompositions of modules over valuation domains

Central to the study of modules is the extent to which decompositions into indecomposable summands are unique. Our investigations into these matters are motivated by the Krull-Schmidt theorem and its connection with local endomorphism rings. Results have been found in the following three settings (1) Finitely generated modules. The main result states that all indecomposable finitely generated modules over break Henselian local rings have local endomorphism rings. This generalizes a result of Swan-Evans from Noetherian to arbitrary Henselian local rings (2) Finite rank torsion free modules. Extending a result of Lady from the very special discrete case, it is shown that finite rank torsion free indecomposable modules over Henselian valuation domains have local endomorphism rings. The discrete valuation domains are precisely the Noetherian valuation domains, thus an extension similar to that found in the finitely generated case is obtained. The proof utilizes a recent result of Dubrovin. A converse is also proved, giving a characterization of Henselian valuation domains. A more satisfying though conditional characterization is realized via a partial extension of Corner's theorem on endomorphism rings of torsion free abelian groups to modules over rank 1 valuation domains of characteristic $\ne$2 (3) Quasi-decompositions and quasi-endomorphism rings. Failing to show that a given indecomposable module has a local endomorphism ring, we study the weaker notion of quasi-decompositions. Let E denote the quasi-endomorphism ring of a finite rank torsion free module M over a commutative domain R. We prove that E is local if and only if M is strongly indecomposable. This gives a result similar to Azumaya's theorem for quasi-decompositions of finite rank torsion free modules, generalizing the result of Reid on abelian groups