Ordered Ring - Basic Properties

Basic Properties

For all a, b and c in R:

• If a ≤ b and 0 ≤ c, then ac ≤ bc. This property is sometimes used to define ordered rings instead of the second property in the definition above.
• |ab| = |a| |b|.
• An ordered ring that is not trivial is infinite.
• Exactly one of the following is true: a is positive, -a is positive, or a = 0. This property follows from the fact that ordered rings are abelian, linearly ordered groups with respect to addition.
• An ordered ring R has no zero divisors if and only if the positive ring elements are closed under multiplication (i.e. if a and b are positive, then so is ab).
• In an ordered ring, no negative element is a square. This is because if a ≠ 0 and a = b2 then b ≠ 0 and a = (-b )2; as either b or -b is positive, a must be positive.