Module-finiteness of Laurent polynomial rings

Posted: March 11, 2010 in Elementary Algebra; Problems & Solutions, Rings and Modules
Tags: , , ,

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

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