Elementary number theory is largely about the ring of integers, denoted by the symbol . The integers are an example of an algebraic structure called an integral domain. This means that satisfies the following axioms:
(a) has operations + (addition) and (multiplication). It is closed under these operations, in that if , then and .
(b) Addition is associative: If , then
(c) There is an additive identity : For all ,
(d) Every element has an additive inverse: If , there is an element such that
(e) Addition is commutative: If , then
(f) Multiplication is associative: If , then
(g) There is an multiplicative identity : For all ,
(h) Multiplication is commutative: If , then
(i) The Distributive Laws hold: If , then
(j) There are no zero divisors: If and , then either or .