- #1

rayman123

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## Homework Statement

Let A be an integral domain with field of fractions K, and suppose that [tex]f\in A[/tex] is non zero and not a unit. Prove that [tex]A[\frac{1}{f}][/tex] is not a finite A-module.

[Hint: if it has a finite set of generators then prove that [tex]1,f^{-1},f^{-2},...,f^{-k}[/tex] is a set of generators for some [tex]k>0[/tex], so that [tex] f^{-(k+1)}[/tex] can be expressed as a linear combination of this. Use this to prove that f is a unit.

## Homework Equations

We assume that [itex]f\neq 0[/tex] and [tex] f\in A\setminus A^{*}[/tex] where I denote [tex]A^{*}[/tex] as the set of units

[tex]A[\frac{1}{f}]=\{p(\frac{1}{f}); p(x)\in A[x]\}[/tex]

## The Attempt at a Solution

Proof by contradiction

suppose that [tex]A[\frac{1}{f}][/tex] is a finite module, that is [tex]A[\frac{1}{f}]=\displaystyle\sum_{i=1}^{n}Ap_{i}[/tex] for [tex]p_{i}(x)\in A[x][/tex]

Let [tex]k=max deg p_{i}(x)[/tex]

[tex]A[\frac{1}{f}]=A\cdot 1+A\cdot f^{-1}+......+A\cdot f^{-k}[/tex]

(how did we get this equations???)

hence [tex]\exists a_{0},......,a_{k}\in A[/tex] (based on what there exists such elements??)

s.t [tex]f^{-(k+1)}=a_{0}+a_{1}f^{-1}+...+a_{k}f^{-k}[/tex] which gives

[tex]f^{-1}=a_{0}f^{k}+a_{1}f^{k-1}+....+a_{k}\in A[/tex] and we get [tex]f\in A^{*}[/tex] hence [tex]A[\frac{1}{f}][/tex] is infinite.

Could someone explain me the second part of the proof and its conslusions?

Thank you