Decomposition to indecomposable projective $A-$modules

Question

I read "Representation theory", written by Alexander Zimmermann. In the chapter 1.11, It is mentioned that:

If $A$ be an Artinian algebra then there is a decomposition $$A=P_1\oplus P_2\oplus \dots\oplus P_m$$ into a direct sum of indecomposable projective $A-$module.

Can someone help me, how do we can prove the statement?

Answer 1

First, any artinian module $M$ is a finite direct sum of indecomposable submodules. To prove this, suppose it is not true and consider the collection of all submodules of $M$ that cannot be written as a finite direct sum of indecomposable submodules. Since $M$ is artinian, this collection (which by hypothesis contains $M$ and so is nonempty) has a minimal element, say $N$. Then $N$ is not indecomposable, so we have $N=N_1\oplus N_2$ for some proper submodules $N_1$ and $N_2$. But by minimality of $N$, each of $N_1$ and $N_2$ is a finite direct sum of indecomposable submodules, and hence so is $N$. This is a contradiction.

Now apply this to $A$ as a module over itself to find that $A$ is a finite direct sum $P_1\oplus \dots\oplus P_m$ of indecomposable submodules. Since $A$ is projective and a direct summand of a projective module is projective, the $P_i$ are all projective.