Let $R[\alpha,~\alpha^{-1}]$ be an extension of a Noetherian integral domain $R$ where $\alpha$ is an element of an algebraic field extension over the quotien field of $R$. In the case $\alpha$ is an anti-integral element over $R$ we will give a condition for a prime ideal $p$ of $R$ to be $pR[\alpha,~\alpha^{-1}] = R[\alpha,~\alpha^{-1}]$. By making use of this we will proceed mainly with the study of flatness and faithful flatness of the extension $R[\alpha,~\alpha^{-1}]/R$. Let $\eta_{1},~\cdots,~\eta_{d}$ be the coefficients of the minimal polynomial of $\alpha$ over the quotient field of $R$. Then we will also investigate the extension $R[\eta_{1},~\cdots,~\eta_{d}]/R$.