Why are the integers a commutative ring with unity but not a field?

The set of integers is a commutative ring with unity because it satisfies the fundamental axioms of addition and multiplication, including associativity, commutativity, and the existence of both additive and multiplicative identities (0 and 1, respectively).12

However, the integers do not constitute a field because they fail to satisfy the tenth field axiom: the existence of multiplicative inverses for all non-zero elements. In a field, every non-zero element $a$ must have a multiplicative inverse $a^{-1}$ such that $a{a}^{-1}=1$. Within the set of integers, only 1 and -1 have multiplicative inverses that are also integers; for any other integer (such as 2), the required inverse (1/2) is not an element of the set. Because the integers lack these multiplicative inverses, they remain an integral domain rather than a field.2