Prove that the elements $b_1=(x_1,y_1)$, $b_2=(x_2,y_2)$ form a base of $Z \oplus Z$ if and only if, $ x_1y_2-y_1x_2=±1 $.
This exercise is from ANEIS E MODULOS ,Francisco Cesar Polcino Milles.
Proof:
Since $b_1,b_2$ are base there exists $a_1,a_2,a_3,a_4$ integers such that
$$ a_1b_1+a_2b_2=(1,0) $$
$$ a_3b_1+a_4b_2=(0,1) $$
Then
$$ a_1x_1+a_2x_2=1 \wedge a_1y_1+a_2y_2=0 ...(1)$$
$$ a_3x_1+a_4x_2=0 \wedge a_3y_1+a_4y_2=1 ...(2)$$
Hence $ (x_1,x_2)=1 \wedge (y_1,y_2)=1$.
Also for (1) $y_1 | a_2y_2 $, therefore $ y_1|a_2$. i.e exist an integer $k$ such that
$ ky_1=a_2$
Replacing in (1)
$$a_1y_1+(ky_1)y_2=0 \rightarrow a_1=-ky_2$$
$$(-ky_2)x_1+(ky_1)x_2=0 \rightarrow -k(x_1y_2 -y_1x_2)=1$$
Hence $(x_1y_2 -y_1x_2) | 1 \rightarrow x_1y_2-y_1x_2=±1$
The converse is simple
Is this demonstration okay?