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 ! 



number of indecomposable summands of an extension of two modules
The answers to both questions are no in general, with a counter-example being given by the path algebra of a quiver of type $\mathsf{D}\_4$—the category of left modules over this algebra is a Hom-finite Krull–Schmidt category.


First, there is an (Auslander–Reiten) short exact sequence


$$0\to\begin{smallmatrix}0\\0&1\\0\end{smallmatrix}\to\begin{smallmatrix}1\\0&1\\0\end{smallmatrix}\oplus\begin{smallmatrix}0\\1&1\\0\end{smallmatrix}\oplus\begin{smallmatrix}0\\0&1\\1\end{smallmatrix}\to\begin{smallmatrix}1\\1&2\\1\end{smallmatrix}\to 0$$


with indecomposable outer terms, but whose middle term is a direct sum of three indecomposables. (Here the notation indicates that vector spaces of the dimensions on the left are being included generically into a vector space of dimension on the right—such things are indecomposable when this list of dimensions is a positive root of $\mathsf{D}\_4$.)


Moreover, there is a $2$-dimensional space of extensions from $\begin{smallmatrix}1\\1&1\\1\end{smallmatrix}$ to $\begin{smallmatrix}0\\0&1\\0\end{smallmatrix}$, where the middle term is generically given by the indecomposable $\begin{smallmatrix}1\\1&2\\1\end{smallmatrix}$, but there are also non-split extensions with decomposable middle term, e.g. $\begin{smallmatrix}1\\0&1\\0\end{smallmatrix}\oplus\begin{smallmatrix}0\\1&1\\1\end{smallmatrix}$.


