basic algebra
Let $A$ be a finite dimensional, unital algebra over a field $k$. By KrullSchmidt Theorem $A$ can be decomposed as a (right) $A$module as follows:
$$A\simeq {P}_{1}\oplus \mathrm{\cdots}\oplus {P}_{k}$$ 
where each ${P}_{i}$ is an indecomposable module^{} and this decomposition is unique.
Definition. The algebra^{} $A$ is called (right) basic if ${P}_{i}$ is not isomorphic^{} to ${P}_{j}$ when $i\ne j$.
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 $A$ over a field $k$ is basic if and only if the algebra $A/\mathrm{rad}A$ is isomorphic to a product^{} of fields $k\times \mathrm{\cdots}\times k$. Thus $A$ is right basic iff it is left basic;

2.
Every simple module over a basic algebra is onedimensional;

3.
For any finitedimensional, unital algebra $A$ over $k$ there exists finitedimensional, unital, basic algebra $B$ over $k$ such that the category^{} of finitedimensional modules over $A$ is $k$linear equivalent^{} to the category of finitedimensional modules over $B$;

4.
Let $A$ be a finitedimensional, basic and connected (i.e. cannot be written as a product of nontrivial algebras) algebra over a field $k$. Then there exists a bound quiver $(Q,I)$ such that $A\simeq kQ/I$;

5.
If $(Q,I)$ is a bound quiver over a field $k$, then both $kQ$ and $kQ/I$ 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
Title  basic algebra 

Canonical name  BasicAlgebra 
Date of creation  20130322 19:17:10 
Last modified on  20130322 19:17:10 
Owner  joking (16130) 
Last modified by  joking (16130) 
Numerical id  5 
Author  joking (16130) 
Entry type  Definition 
Classification  msc 13B99 
Classification  msc 20C99 
Classification  msc 16S99 