Ordered ring

In abstract algebra, an ordered ring is a special kind of mathematical structure. It is usually a commutative ring called R, which has a way to compare numbers called a total order ≤. This means we can say if one number is less than, equal to, or greater than another number.

There are two important rules in an ordered ring. First, if a is less than or equal to b, then adding the same number c to both a and b keeps the order the same. In other words, a + c will also be less than or equal to b + c.

The second rule deals with multiplying numbers. If both a and b are greater than or equal to zero, then multiplying them together also gives a number that is greater than or equal to zero. This helps us understand how positive numbers behave when we multiply them together.

Ordered rings are important in mathematics because they let us study numbers and their relationships in a structured way. They combine the ideas of rings, which are systems for adding and multiplying, with the idea of ordering numbers, which helps us understand their size and relationship to each other.

Examples

Ordered rings are easy to understand because we see them in everyday math. Examples include the integers, the rationals, and the real numbers. The rationals and reals are special types called ordered fields. But the complex numbers are different because we cannot say if 1 is bigger or smaller than i.

Positive elements

In ordered rings, we call an element positive if it is greater than zero. This is similar to how we think about positive numbers with real numbers, like 1 or 5. Some areas of study use special symbols to talk about all nonnegative elements (zero and positive) and all positive elements separately.

Absolute value

If a number a is part of an ordered ring, we can find its absolute value. This is written as |a|.

The absolute value of a tells us how far a is from zero, no matter if it’s positive or negative. It’s like measuring distance without caring about direction.

Discrete ordered rings

A discrete ordered ring is a special kind of math system where we can compare numbers, and there are no numbers in between 0 and 1. The whole numbers, like 0, 1, 2, and so on, are an example of a discrete ordered ring. However, the fractions, or rational numbers, are not a discrete ordered ring because you can always find another number between any two fractions.

Basic properties

In an ordered ring, certain rules help us understand how numbers work together. For example, if you have numbers a, b, and c where a is less than or equal to b and c is zero or more, then multiplying a by c will give a result that is less than or equal to multiplying b by c. Also, the absolute value of a product of two numbers is the same as the product of their absolute values.

Ordered rings that are not simple (or "trivial") always have infinitely many numbers. For any number a, exactly one of these is true: a is positive, the negative of a is positive, or a is zero. Additionally, in an ordered ring, a negative number can never be a square of another number. This is because squares are always zero or positive.