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 .