## Module-finiteness of Laurent polynomial rings

Posted: March 11, 2010 in Elementary Algebra; Problems & Solutions, Rings and Modules
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Let $R$ be a commutative ring with identity and $S=R[x,x^{-1}],$ the ring of Laurent polynomials with coefficients in $R.$ Obviously $S$ is not a finitely generated $R$-module but we can prove this:

Problem. There exists $f \in S$ such that $S$ is a finitely generated $R[f]$-module.

Solution. Let $f=x+x^{-1}.$ Then $x=f - x^{-1}$ and $x^{-1}=f-x.$ Now an easy induction shows that $x^n \in xR[f]+R[f]$ for all $n \in \mathbb{Z}.$ Hence $S=xR[f] + R[f]. \ \Box$