We define Gröbner bases for submodules of $\mathbb Z^n$ and
characterize minimal and reduced bases combinatorially in terms
of minimal elements of suitable partially ordered subsets of $\mathbb Z^n$.
Then we show that Gröbner bases for saturated pure binomial
ideals of K[x_1, . . . , x_n], char (K) ≠ 2, can be immediately derived
from Gröbner bases for appropriate corresponding submodules
of $\mathbb Z^n$. This suggests the possibility of calculating the Gröbner
bases of the ideals without using the Buchberger algorithm.
