basic algebra

Let $A$ be a finite dimensional, unital algebra over a field $k$. By Krull-Schmidt 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 one-dimensional;

3. For any finite-dimensional, unital algebra $A$ over $k$ there exists finite-dimensional, unital, basic algebra $B$ over $k$ such that the category of finite-dimensional modules over $A$ is $k$-linear equivalent to the category of finite-dimensional modules over $B$;

4. Let $A$ be a finite-dimensional, 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.