Consider the following pushout diagram $\require{AMScd}$ \begin{CD} A @>f>> B\\ @V g V V @VV V\\ C @>> > D \end{CD} in the category of $\textbf{Rings}$ where $f,g$ both are flat monomorphisms. The diagram is also a pullback diagram.

I'm trying to show that it's a pullback diagram but could not.