In mathematics, a ring is a structure consisting of a set `G` together with two binary operations + and · with the following properties:

- (
`G`, +) is an Abelian group - · is associative on
`G` - · is both left and right distributive with respect to + on
`G`.

Note that · does not need to be commutative. If · is commutative, then the ring is referred to as a commutative ring.

Perhaps the simplest example of a ring is (`I`, +, ·); in other words, the set of integers with ordinary addition and multiplication.

A somewhat different example of a ring is as follows: Let `A` = {0, 1, 2, 3, 4}. Let `a` ⨁ `b` represent the remainder obtained by dividing `a` + `b` by 5, and `a` ⨂ `b` represent the remainder obtained by dividing `ab` by 5. By enumerating the possible values of the two operations, it can be shown that (`A`, ⨁, ⨂) is a ring.

When reading about rings in various books, authors will differ as to whether a ring must have an identity element for the second operation of a ring. Going back to our first example of the set of integers together with addition and multiplication, "1" is an identity element for the second operation of this ring. So, more precise terminology is often used. For example, a ring with an identity element for the second operation might be referred to as a *ring with unity* or *ring with identity*, while a ring without such an identity element might be referred to as a *pseudo-ring*.

See also: laws of arithmetic.

Sources used (see bibliography page for titles corresponding to numbers): 51.