ordered ring

If 0, and x is strictly positive if x>0.

The set of all positive elements of an ordered ring A is denoted by A_+, and the set of all strictly positive elements of A is denoted by A^*₊.

If (A,+,·,<) is an ordered ring and if < is a total ordering, we shall, of course, call (A,+,·,<) a totally ordered ring; if (A,+,·) is a field, we shall call (A,+,·,<) an ordered field, and if, moreover, < is a total ordering, we shal call (A,+·,<) a totally ordered field.

The positive elements in an ordered ring allow us to compare elements to 0, but we know in the integers that we can compare any two elements to each other. For example, we know that 4>2 because 4-2>0. We can extend this idea to any ordered ring. If R is an ordered ring and a,b∈R, then we know by trichotomy that exactly one of the following must be true: a-b>0, a-b=0, or -(a-b)>0.