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Real exponentiation in the quotients of rings of continuous functions by prime ideals

Fix prime ideal $I$, $r\in\mathbb{R}_{>0}$ and $a = f+I = g+I$ with $f, g\geq 0$. I'll prove that $f^r+I = g^r+I$. First, note if $r\in\mathbb{R}_{>0}$ and $a = 0$, then $f^r = f^{\frac{1}{n}}f^{...
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