Let $k$ be a field, $A$ a finite dimensional $k$-algebra, $X$ a finite dimensional indecomposable (left) $A$-module and $M$ an infinite dimensional (left) $A$-module. Further $X\subseteq M$ and for every finite dimensional $A$-module $N$ with $X\subseteq N\subseteq M$ the inclusion $X\hookrightarrow N$ splits. Is it true that $X \hookrightarrow M$ must split? If not, does somebody know a counterexample?



Split monomorphisms of modules - does the finite case imply the infinite case?
Since $A$ is a finite-dimensional $k$-algebra, $M$ is the direct limit of its finite-dimensional submodules $N$ which contain $X$. Now the sequence $0\to X\to M\to M/X\to 0$ is pure-exact, since it is the direct limit of the split exact sequences $0\to X\to N\to N/X\to 0$. Also $X$ is finite-dimensional over $k$, so it is pure-injective. Thus the sequence splits, and $X$ is a direct summand of $M$.


