Let $Mat_2(\mathbb{Z})$ be the $\mathbb{Z}$-algebra of $2\times2$ matrices with integer entries. Let $A$ be a $\mathbb{Z}$-submodule of $Mat_2(\mathbb{Z})$ containing $\mathbb{Z}$. We want to show that $A$ is a subring, does it suffice to show that the product of two elements in $A$ lies in $A\otimes\overline{\mathbb{Q}}$?
Seen that if $A$ is the $\mathbb{Z}$-submodule $\mathbb{Z}\oplus 2\mathbb{Z}\oplus0\oplus0$ of $B=\mathbb{Z}\oplus\mathbb{Z}\oplus\mathbb{Z}\oplus\mathbb{Z}$, the equality $(A\otimes\overline{\mathbb{Q}})\cap B=A$ does not hold.
I do not understand how does it work the argument of the notes of B. Conrad http://math.stanford.edu/~conrad/676Page/handouts/picgroup.pdf page 4, the 14th-16th lines of section 3.