Hi everyone.

I am working with lattices in $\mathbb{C}$, and I want to know whether a certain vector is an element of the lattice.

In particular, suppose my lattice vectors are $a$ and $b$ and I want to know whether $c$ (a Gaussian or Eisenstein integer) is in the lattice. The problem reduces to existence of rational integer solutions to the linear diophantine equation $ax+by=c$. Is there any known conditions to determine whether the Gaussian/Eisenstein integer solutions obtained (usual Bezout's) are rational integers?

Any help will be appreciated. Thanks!