Let $R$ be a PID , let $a,b \in R$ be co-prime elements , i.e. $aR+bR=R$ . Then obviously $R[X]/(aX-b)$ is an integral domain . How to show that $R[X]/(aX-b) \cong R[1/a]$ ?
2026-03-30 17:35:40.1774892140
$a,b$ co-prime elements in PID $R$ , then to show $R[X]/(aX-b) \cong R[1/a]$
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Hint:
Consider a Bézout's relation: $\;ua+vb=1,\quad (u,v\in R)$. In $R[1/a]$, you have: $$\frac 1a=u+\frac{vb}a.$$