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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.

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 ratios. 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.

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.

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.

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. A ring is ordered if zero is neither positive nor negative.

The integers is the unique minimal ordered ring.

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.