The socle of cokernel of irreducible monomorphisms in the AR quiver of type An is simple.

I believe that this result is hidden in a more general result in some articles, I tried to find a lot but failed.

The following lemma maybe useful:

M be a module of length n. TFAE:

1 SocM is simple

2 Any non-zero submodule of M is indecomposable

3 There exists a composition series of M with all terms indecomposable

Any suggestions are welcome!

The socle of cokernel of irreducible monomorphisms in the AR quiver of type An/I is simple.

I believe that this result is hidden in a more general result in some articles, I tried to find a lot but failed.

The following lemma maybe useful:

M be a module of length n. TFAE:

1 SocM is simple

2 Any non-zero submodule of M is indecomposable

3 There exists a composition series of M with all terms indecomposable

Any suggestions are welcome!