Let $Ax=B$ be a system of linear diophantine equations, where $A$ is a full rank $n \times 2n$-matrix with integer entries. In the case $n=1$ we have solutions parameterized by $\mathbb{Z}$ iff $gcd(a_{11},a_{12})$ divides $b_{11}$. Is there a similar statement for arbitrary $n$ of the form "We get solutions paramterized by $\mathbb{Z}^n$ iff *condition on $B$*"

EDIT: Since I know basically nothing about diophantine equations, it could be that this question is far from being research level.. if so, just tell me and I will delete it and bring it up in math SE