# 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:

 $A\simeq P_{1}\oplus\cdots\oplus P_{k}$

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 $i\neq j$.

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]):

1. 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 $k\times\cdots\times k$. Thus $A$ is right basic iff it is left basic;

2. 2.

Every simple module over a basic algebra is one-dimensional;

3. 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. 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. 5.

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

## References

Title basic algebra BasicAlgebra 2013-03-22 19:17:10 2013-03-22 19:17:10 joking (16130) joking (16130) 5 joking (16130) Definition msc 13B99 msc 20C99 msc 16S99