We consider finitely generated modules over an Artin algebra. Let $X$ be an indecomposable module and let $f:X \longrightarrow X \oplus X$ a monomorphism. Must $f$ always be a split monomorphism?



Must the inclusion of an indecomposable module in the direct sum of two copies always split?
Yes, it must be split.


Since $M$ is an indecomposable module for an Artin algebra, its endomorphism ring $E$ is a local ring with nilpotent Jacobson radical $J(E)$. Say $J(E)^n=0$.


Let the monomorphism $\varphi:M\to M\oplus M$ be given by $\varphi(m)=(\alpha(m), \beta(m))$. If either $\alpha$ or $\beta$ is an isomorphism, then $\varphi$ splits, so we shall assume that $\alpha,\beta\in J(E)$.


Consider the sequence of monomorphisms
$$M\xrightarrow{\varphi}M\oplus M\xrightarrow{(\varphi,\varphi)}M\oplus M\oplus M\oplus M\xrightarrow{(\varphi,\varphi,\varphi,\varphi)}\cdots.$$


Since $J(E)^n=0$, the composition of the first $n$ maps in the sequence is zero, contradicting the fact that they are all monomorphisms.


