You are here
Home ›basic algebra
Primary tabs
basic algebra
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.
Definition. The algebra is called (right) basic if is not isomorphic to when .
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 [1]):
Proposition.
1. A finite algebra over a field is basic if and only if the algebra is isomorphic to a product of fields . Thus is right basic iff it is left basic;
2. Every simple module over a basic algebra is one-dimensional;
3. For any finite-dimensional, unital algebra over there exists finite-dimensional, unital, basic algebra over such that the category of finite-dimensional modules over is -linear equivalent to the category of finite-dimensional modules over ;
4. 5. If is a bound quiver over a field , then both and are basic algebras.
References
- 1 I. Assem, D. Simson, A. Skowronski, Elements of the Representation Theory of Associative Algebras, vol 1., Cambridge University Press 2006, 2007
Mathematics Subject Classification
13B99 None of the above, but in MSC2010 section 13Bxx20C99 None of the above, but in MSC2010 section 20Cxx
16S99 None of the above, but in MSC2010 section 16Sxx
- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)
- Other useful stuff
Recent Activity
new image: sinx_approx.png by jeremyboden
new image: approximation_to_sinx by jeremyboden
new image: approximation_to_sinx by jeremyboden
new question: Solving the word problem for isomorphic groups by mairiwalker
new image: LineDiagrams.jpg by m759
new image: ProjPoints.jpg by m759
new image: AbstrExample3.jpg by m759
new image: four-diamond_figure.jpg by m759
May 16
new problem: Curve fitting using the Exchange Algorithm. by jeremyboden
new question: Undirected graphs and their Chromatic Number by Serchinnho


