Invertibility of elements in a left Noetherian ring

Recall that we get an isomorphism $A^\text{op} \to \operatorname{End}_A(A)$ by sending $a$ to the endomorphism $x \mapsto xa$. Here $A^\text{op}$ is the opposite ring. If $a$ is left invertible then the corresponding endomorphism $f$ is surjective, and if we can show that $f$ is injective then $f$ is invertible in $\operatorname{End}_A(A)$, whence $a$ is invertible in both $A^\text{op}$ and $A$.

It isn't any harder to prove a more general statement: If an endomorphism of a Noetherian module is surjective, then it is an isomorphism.

Here are some ideas for this. If $g$ is such an endomorphism then the increasing sequence of submodules $\{\operatorname{Ker}(g^n)\}$ must stabilize. Use this and the fact that each $g^n$ is surjective to show that the kernel is trivial.