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?
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1$\begingroup$ If $R = R_1 \times \cdots \times R_n$ is the decomposition into local Artinian rings, then $X$ can only be supported on one of them, so you can assume $R$ is local. $\endgroup$– R. van Dobben de BruynCommented Mar 14, 2021 at 16:33
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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.
answered Mar 14, 2021 at 18:56
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$\begingroup$ Thanks for your answer! You probably mean $\alpha^n$ instead of $\varphi^n$. So that means that the intersection of two kernals of nilpotent maps must always be non-zero. A similar argument would work for an arbitrary number of nilpotent maps. One could take the interesection of all kernals of nilpotent maps. Do you have an idea what module one would get like this? $\endgroup$ Commented Mar 14, 2021 at 19:05
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$\begingroup$ Thanks. Actually, I meant $J(E)^n=0$ (fixed now): I need every composition of $n$ maps, each of which is either $\alpha$ or $\beta$, to be zero. $\endgroup$ Commented Mar 14, 2021 at 19:11
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$\begingroup$ @kevkev1695 This is the socle of $M$, when considered as an $\mathrm{End}(M)$ module. In small examples, it seems to also be the socle as an $R$-module, but I don't know if that is generally right. $\endgroup$ Commented Mar 17, 2021 at 14:06