= Noetherianity of subalgebras of a one-variable Laurent polynomial ring
{c}
Every unital $k$-<subalgebra> $A\subseteq k[T,T^{-1}]$ is a <Noetherian ring>. If $A\subseteq k[T]$ or $k[T^{-1}]$, a nonconstant element $f\in A$ makes that ambient <polynomial ring> a <finitely generated module> over $k[f]$. Otherwise $A$ contains an $f$ with both positive and negative exponents, and both $T$ and $T^{-1}$ are <integral elements> over $k[f]$. In either case $A$ is a <submodule> of a finite module over the <Noetherian ring> $k[f]$, and every <ideal> of $A$ is finitely generated over $k[f]$, hence over $A$.
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