Let $R$ be a ring, $S$ be a subring of $R$ and $x$ be an invertible element of $R$. I am trying to show all elements in the intersection of the two subrings generated by $x$ and its inverse (with $S$) is integral over $S$. I am considering if the $S$-module is closed under multiplication by $y$, then anything raised by its power is in $M$. Then perhaps forming some polynomials $f(x)$ and $g(1/x)$ would be a good start here but I am not sure how to proceed..Say let $F=a_nX^n +..+ a_0, G=b_mX^m+..+b_0$ ?
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1See Theorem 21 in my A few facts on integrality (version 6) for a more general fact. I don't fully remember how much of my note you need to read for this proof, but I believe it was just the first few definitions in §1 and §5; everything else is irrelevant. – darij grinberg Mar 26 '17 at 00:42
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It does not look like the approach i am looking for but I'll look more into it to try understand. – Homaniac Mar 26 '17 at 01:06