Ring
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A ring is a structure of abstract algebra, similar to a group or a field. A ring is a set of elements with two operations, usually called multiplication and addition and denoted
and
, which have the following properties:
Under the operation +, the ring is an abelian group and so obeys all the group axioms (existence of an identity, existence of inverses, associativity) as well as commutivity.
There exists an element, usually denoted 1, such that for all
. (Multiplicative identity.)
For every three elements we have
. (Associativity.)
For every three elements we have
and
. (Distributivity of multiplication over addition.)
Note especially that multiplicative inverses need not exist, and that multiplication need not be commutative.
Common examples of rings include the integers or the integers taken modulo , with addition and multiplication as usual. In addition, every field is a ring.