Quotient of a quotient.

Question

Let $\mathfrak{g}$ be a complex reductive Lie algebra. Let $M,N,L$ be finite dimensional $\mathfrak{g}$-modules.

Suppose $M$ is a quotient of $N$ and $N$ is a quotient of $L$.

My question: Is $M$ a quotient of $L$?

Answer 1

If $q: L \twoheadrightarrow M$ is the canonical quotient map for $M$, and $p: M \twoheadrightarrow N$ is the canonical quotient map, then $pq: L \to N$ is surjective, and thus $N \cong L / \mathrm{ker}(pq)$.

Note that this did not use any structure of $\mathfrak{g}$ or the finite-dimensionality of the modules $L$, $M$, and $N$; it follows only from the fact that if $\varphi: A \twoheadrightarrow B$ is a surjective morphism of "modules," then $B \cong A / \mathrm{ker}(\varphi)$. This is true for abelian groups, and a "module" of any kind should be an abelian group with extra structure. In fact, this fact is true* in any abelian category, which are the most general categories of modules.

*: modulo^{wink wink} replacing surjectivity with the categorical notion of epimorphism.