I have the following question : in a Krull-Schmidt category (say the category of finite length left modules over a ring, this is the case which interests me), is it possible to relate the number of indecomposable summands of an extension $E$ of $M$ by $N$ with the number of indecomposables of $M$ and $N$ ?

More precisely, if we write $M=M_1\oplus...\oplus M_r $ and $N=N_1\oplus ...\oplus N_s$ where all the $M_i$ and $N_i$ are indecomposable, and given an exact sequence $0\longrightarrow N \longrightarrow E\longrightarrow M\longrightarrow 0$ and $E=E_1\oplus...\oplus E_q$ with $E_i$ indecomposable again, do we have $q\leq r+s$ ?

I am interested in the equality case too. In case we have $q=r+s$, is $E$ isomorphic to the direct sum $M\oplus N$ ?

I would like to tell you that in case of $\mathbf{Z}$-modules of finite length, all the answers are positive :just consider good bases for the inclusion $N\longrightarrow E$ to see that it is sufficient to treat the case where $M,E,N$ are cyclic.

Thank you !

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