Skew Hilbert basis theorem ($A\text{ left Noetherian}⟹A[t;\sigma ]\text{ left Noetherian}$)

For an automorphism $\sigma$ of a left Noetherian ring $A$, the skew polynomial ring of an automorphism $A[t;\sigma ]$ is left Noetherian. For a left ideal $L$, the normalized leading-coefficient ideals $I_n=\{{\sigma }^{-n}(a_n):{\sum }_{j=0}^n{a}_j{t}^j\in L\}$ form an ascending chain of left ideals of $A$. Choose finite generating lifts through the stabilization degree. Every element of $L$ has its leading coefficient cancelled by a suitable left multiple of one of these lifts, and induction on degree proves finite generation. Passing to the skew Laurent polynomial ring preserves the property by clearing negative powers. The right-sided statement follows by applying the argument to the opposite ring.