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Let  R  be a ring and  Q  be a finite quiver, and let    be the number of vertices of  Q . Let    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    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    representations in    is unique up to  n  permutations of  m  elements. Let   . A non-directed graph    associated to  M  is introduced and is shown to determine the unique decomposition of  M  into indecomposable representations. This class of representations    is shown to generalize the known classes of modules for which a theorem analogous to the    case of our main theorem holds.

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