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. 

Similarly, the positive numbers in a field are the sums of ones or their reciprocals. A field is minimal if every number is positive, negative, or zero. 

$\bf Q$ is the unique minimal field in which zero is neither positive nor negative.

${\bf Z}/p\bf Z$ is the unique minimal field in which zero is either positive or negative.