= Skew Hilbert basis theorem
{title2=$A\text{ left Noetherian}\implies A[t;\sigma]\text{ left Noetherian}$}
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 $I_n=\{\sigma^{-n}(a_n):\sum_{j=0}^n a_jt^j\in L\}$ 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>.
Back to article page
