Lattice formed by a linear congruence

Question

Let there be a linear congruence $a+b y \equiv 0 \pmod{m}$, with $y$ and $m$ ($m$ is a prime) values known. Do all the integer $(a,b)$ pairs satisfying the congruence form a lattice? If yes, how can I find two vectors (i.e. the basis) which generate the lattice mentioned?

Answer 1

Hint $\rm\,\ mod\ m\!:\ a\equiv -by \iff (a,b) = (km-ny,n)\, =\, k(m,0) + n(-y,1)$