Representations of quivers over a ring and the Weak Krull–Schmidt Theorems
-
Nicola Girardi
Abstract.
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.
© 2012 by Walter de Gruyter Berlin Boston
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- Representations of quivers over a ring and the Weak Krull–Schmidt Theorems
- Logarithmically improved regularity criteria for the 3D viscous MHD equations
- Tilting modules and universal localization
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- Hardy's inequality in the scope of Dirichlet forms
- On some Banach spaces of martingales associated with function spaces
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Articles in the same Issue
- Masthead
- Representations of quivers over a ring and the Weak Krull–Schmidt Theorems
- Logarithmically improved regularity criteria for the 3D viscous MHD equations
- Tilting modules and universal localization
- A bound on the degree of schemes defined by quadratic equations
- Hardy's inequality in the scope of Dirichlet forms
- On some Banach spaces of martingales associated with function spaces
- Construction of quasi-periodic response solutions in forced strongly dissipative systems
- On the description of Leibniz superalgebras of nilindex
- Presentations of graph braid groups
- On the vanishing of the lower K-theory of the holomorph of a free group on two generators
- Boundedly generated subgroups of finite groups
