Let $R\subset T \subset F_R$, where $R,T$ are two integral domains and $F_R$ is the quotient field of $R$. I need to show that $F_T\cong F_R$.
My effort: Since $T$ embeds in a field $F_R$, it must contain an isomorphic copy of its quotient field, i.e $T\subset F_T\subset F_R$. Since $R$ is contained in the field $F_T$, we must conclude that an isomorphic copy of $F_R$ is contained in $F_T$.
Therefore we conclude that $R\subset T\subset F_R'\subset F_T\subset F_R$, where $F_R'$ is an isomorphic copy of $F_R$ inside itself. I feel I am very close to proving $F_T\cong F_R$, but lack the closing argument. Any help is appreciated.