This has been beaten to death, but here goes...
The positive numbers in a ring are the sums of ones, i.e. the set of positive numbers is the smallest inductive subset of the ring. The negative numbers are the negatives of the positive numbers. A ring is minimal if every number is positive, negative, or zero.
$\bf Z$ is the unique minimal ring in which zero is neither positive nor negative.
${\bf Z}/n\bf Z$ is the unique minimal ring in which zero is either positive or negative.