Let be a finite dimensional, unital algebra over a field . By Krull-Schmidt Theorem can be decomposed as a (right) -module as follows:
where each is an indecomposable module and this decomposition is unique.
Of course we may easily define what does it mean for algebra to be left basic. Fortunetly these properties coincide. Let as state some known facts (originally can be found in ):
Let be a finite-dimensional, basic and connected (i.e. cannot be written as a product of nontrivial algebras) algebra over a field . Then there exists a bound quiver such that ;
If is a bound quiver over a field , then both and are basic algebras.
|Date of creation||2013-03-22 19:17:10|
|Last modified on||2013-03-22 19:17:10|
|Last modified by||joking (16130)|