The "'Auslander-Reiten quiver "'of an Artin algebra has a vertex for each indecomposable module and an arrow between vertices if there is an irreducible morphism between the corresponding modules.
12.
The Krull Schmidt theorem and related results give conditions under which a module has something like a primary decomposition, a decomposition as a direct sum of indecomposable modules in which the summands are unique up to order.
13.
If the ring " R " is semiperfect, then " R " is a direct sum of principal indecomposable modules, and there is one isomorphism class of PIM per isomorphism class of simple module.
14.
Then there are only finitely many isomorphism types of indecomposable modules in the block, and the structure of the block is by now well understood, by virtue of work of Brauer, E . C . Dade, J . A . Green and J . G . Thompson, among others.
15.
Over semisimple rings, this is no loss as every module is a semisimple module and so a quivers ( whose nodes are the simple modules and whose edges are composition series of non-semisimple modules of length 2 ) and Auslander Reiten theory where the associated graph has a vertex for every indecomposable module.
16.
Over a Dedekind domain that is not a PID, the ideal class group measures the difference between projective indecomposable modules and principal indecomposable modules : the projective indecomposable modules are exactly the ( modules isomorphic to ) nonzero ideals and the principal indecomposable modules are precisely the ( modules isomorphic to ) nonzero principal ideals.
17.
Over a Dedekind domain that is not a PID, the ideal class group measures the difference between projective indecomposable modules and principal indecomposable modules : the projective indecomposable modules are exactly the ( modules isomorphic to ) nonzero ideals and the principal indecomposable modules are precisely the ( modules isomorphic to ) nonzero principal ideals.
18.
Over a Dedekind domain that is not a PID, the ideal class group measures the difference between projective indecomposable modules and principal indecomposable modules : the projective indecomposable modules are exactly the ( modules isomorphic to ) nonzero ideals and the principal indecomposable modules are precisely the ( modules isomorphic to ) nonzero principal ideals.
19.
Over a Dedekind domain that is not a PID, the ideal class group measures the difference between projective indecomposable modules and principal indecomposable modules : the projective indecomposable modules are exactly the ( modules isomorphic to ) nonzero ideals and the principal indecomposable modules are precisely the ( modules isomorphic to ) nonzero principal ideals.
20.
The quasi-tilted algebras over " k " are precisely the finite-dimensional algebras over " k " of global dimension & le; 2 such that every indecomposable module either has projective dimension & le; 1 or injective dimension & le; 1 . classified the hereditary abelian categories that can appear in the above construction.
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