Faithful flatness is preserved by isomorphism?

Question

Let $D$ be a ring and let $M$ and $N$ be two modules over $D$.
Suppose that $M$ is a faithfully flat $D-$module and $M$ and $N$ are isomorphic. Prove or disprove that $N$ is a faithfully flat $D-$module?

Answer 1

Hint:

Given an injective linear map $\;i:E\longrightarrow F$, and the isomorphism $\;u:M\longrightarrow N$, consider the commutative diagram: \begin{alignat}{2}\DeclareMathOperator{\id}{id} E\otimes_D M&\xrightarrow{~i\,\otimes\,\id_M~}{}F\otimes_D M \\ \id_E\otimes f\downarrow\qquad & \hspace{5em}\downarrow \id_F\otimes u\\[-1ex] E\otimes_D N&\xrightarrow{~i\,\otimes\,\id_N~}F\otimes_D N \end{alignat}