Consider the $\mathbb Z$-module $\mathbb Z \oplus \mathbb Z$.
I have seen two assertions about the direct summands of $\mathbb Z \oplus \mathbb Z$ but I have trouble with these:
$(a,b)\in\mathbb Z \oplus \mathbb Z$ spans a direct summand if and only if it is primitive, that is, $(a,b)=1$.
Two linearly independent vectors $(a,b)$ and $(c,d)$ span a direct summand of $\mathbb Z \oplus \mathbb Z$ if and only if the determinant of the matrix $\left( \begin{array}{cc} a & c\\b & d \end{array}\right)$ is $±1$.