Definition 2

A ring $\mathbf{A}$ is called an integral domain (or a domain of integrity) if it is commutative, non-zero, and the product of two non-zero elements of $\mathbf{A}$ is non-zero.

The ring $\mathbf{Z}$ of rational integers is an integral domain; it is commutative and non-zero; the product of two integers $> 0$ is non-zero; every non-zero integer is of the form $a$ or $-a$ with $a > 0$ and $(-a)b = -ab,\ (-a)(-b) = ab$ for $a > 0,\ b > 0$, whence our assertion.

Every commutative field is an integral domain. A subring of an integral domain is an integral domain. In particular, a subring of a commutative field is an integral domain. We shall show that conversely every integral domain $\mathbf{A}$ is isomorphic to a subring of a commutative field. Recall (§ 8, no. 12) that $\mathbf{A}$ has been identified with a subring of its total ring of fractions. Our assertion then follows from the following proposition: