I am having a difficult time understanding the proof of a corollary to Cayley-Hamilton theorem in Eisenbud's Commutative Algebra.
The statement is:
Let $R$ be a ring, and let $M$ be a finitely generated $R$-module. If $\alpha\colon M\to M$ is an epimorphism of $R$-modules, then $\alpha$ is an isomorphism.
For the proof, we give $M$ an $R[t]$-module structure, by letting $t\cdot m=\alpha(m)$ for each $m\in M$. Then set $I=(t)$, giving us $IM=M$. Thus we apply Cayley-Hamilton, yielding a monic polynomial $q$ such that $q(id_M)=0$ as an endomorphism. The next line I am having trouble seeing,
It follows that $(id_M-q(t)t)M=0$, or equivalently $id_M-q(\alpha)\alpha=0$.
It's clear $q(t)tM=q(t)M$, but why is $q(t)M=M$?