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

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!

• Do you assume your $A_n$-quiver to be linearly oriented? May 18 '16 at 9:36
• In fact here the orientation and admissable ideal I are all arbitrary. @Julian Kuelshammer May 19 '16 at 5:34
• Every non-projective indecomposable module is the cokernel of an irreducible monomorphism, namely the left almost split morphism in the AR-sequence. I think you mean irreducible monomorphisms between indecomposable modules. May 19 '16 at 10:15