Fraction field of (A[b])

Question

Let $A \subseteq B$ be commutative rings and $b \in B$. Is it true that $Frac(A[b])=Frac(A)[b] ?$

I felt it was true that $Frac(\Bbb Z[i])=\Bbb Q[i]$ so I thought it was maybe true for other rings too but not completely sure about it.

Answer 1

No. Consider $\Bbb{R} \subseteq \Bbb{R}[x],$ then $Frac(\Bbb{R}[x]) = \Bbb{R}(x),$ and $Frac(\Bbb{R})[x] = \Bbb{R}[x].$ Since $\frac{1}{x} \in \Bbb{R}(x)$ and $\frac{1}{x}\notin \Bbb{R}[x],$ then it is clear that $Frac(A[b])\ne Frac(A)[b],$ in general.