Representations of quivers over a ring and the Weak Krull-Schmidt Theorems

Let R be a ring and Q be a finite quiver, and let n >= 1 be the number of vertices of Q. Let T be the class of representations of Q by right R-modules with local endomorphism ring and R-module homomorphisms. The endomorphism ring of a representation M is an element of T has at most n maximal right ideals, all of which are also left ideals, and the isomorphism class of M is determined by n invariants. The main theorem of this paper states that a finite direct sum of m >= 1 representations in T is unique up to n permutations of m elements. Let M is an element of T. A non-directed graph G(M) associated to M is introduced and is shown to determine the unique decomposition of M into indecomposable representations. This class of representations T is shown to generalize the known classes of modules for which a theorem analogous to the n = 2 case of our main theorem holds.