A commutative ring R with identity, with the additional property that

(The axiom numbering follows on from that used for ring.) Thus an integral domain is a commutative ring with identity with no zero-divisors. The natural example is the set ℤ of all integers with the usual addition and multiplication. Any field is an integral domain. Further examples of integral domains (these are not fields) are: the set $\mathbb{Z}[\sqrt{2}]$ of all real numbers of the form $a+b\sqrt{2}$, where a and b are integers, and the set ℝ[x] of all polynomials in a variable x, with real coefficients, each with the normal addition and multiplication.