Hello,

I am sorry that I did not formulate the question precisely enough. The claim I wanted to be true is:

Let $\mathcal{A}$ be an abelian category, in which every object has finite length, and there are enough projectives. Then, there is a bijective correspondence between indecomposable projectives and irreducible objects.

The only compound which I did not know how to proof was:

An indecomposable projective object of finite length has an unique irreducible quotient.

Now, I saw elements which lead to a proof in those notes:

http://math.berkeley.edu/~serganov/math252/notes9.pdf

Here is a proof:

Lemma 1: Every endomorphism of an indecomposable object of finite length is either nilpotent, or isomorphism.

Lemma 2: If the sum of two endomorphisms of an indecomposable object of finite length is an isomorphism, one of them is also an isomorphism.

In the above notes there are the (short and easy) proves of these two lemmas.
Now, let $P$ be an indecomposable projective of finite length. We need to show that it has a unique maximal proper sub-object (thus unique irreducible quotient). Suppose not: let $K_1, K_2 \subset P$ be two different maximal proper sub-objects. Then we have a surjection $K_1 \oplus K_2 \to P$, which splits since $P$ is projective. Hence we get two maps $P \to K_1, P \to K_2$, which when considered as maps $P \to P$, sum to the identity map. Thus, by lemma 2 above, one of them is an isomorphism, which is impossible by length consideration.