**Rings and Fields**

Groups just have one binary operation. To fully explain the behaviour of the different types of numbers, structures with two operators need to be studied. The most important of these are rings, and fields.

A **ring** has two binary operations (+) and (×), with × distributive over +. Under the first operator (+) it forms an *abelian group*. Under the second operator (×) it is associative, but it does not need to have identity, or inverse, so division is not required. The additive (+) identity element is written as 0 and the additive inverse of *a* is written as −*a*.

**Distributivity** generalises the *distributive law* for numbers, and specifies the order in which the operators should be applied, (called the precedence). For the integers (*a* + *b*) × *c* = *a* × *c* + *b* × *c* and *c* × (*a* + *b*) = *c* × *a* + *c* × *b*, and × is said to be *distributive* over +.

The integers are an example of a ring. The integers have additional properties which make it an **integral domain**.

A **field** is a *ring* with the additional property that all the elements excluding 0 form an *abelian group* under ×. The multiplicative (×) identity is written as 1 and the multiplicative inverse of *a* is written as *a*−1.

The rational numbers, the real numbers and the complex numbers are all examples of fields.

Read more about this topic: Algebra, Abstract Algebra

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