Ring Axioms
Definition. A ring is a set
with the operations of addition and multiplication.
Addition
For any
there is an element
, so that R is closed under addition.
Addition is associative, i.e. 
for all
There is an element of
called the zero element and written 0, which has the property that
for all
so that 0 is the identity element of the addition operation.
Every elementhas a negative, an element ofwritten
which satisfies
so that the additive inverse of
is
These together imply that a ring is a group under addition with identity 0.
Addition is commutative, i.e.
for all
This implies that a ring is an Abelian group under addition.
Multiplication
For any
there is an element
Multiplication is associative, i.e. 
for all
Multiplication is distributive over addition, i.e.
and
for all
A ring may or may not have a multiplicative identity. This means that the elements of the ring with the operation of multiplication may not necessarily be a group. Neither must the multiplication operation be commutative.
Examples:
are all rings but the set of all ordered pairs
with first component equal to 1 is not since
and the first component is not 1 so the operation of addition is not closed.
