Integral Domain - Examples

Examples

• The prototypical example is the ring Z of all integers.
• Every field is an integral domain. Conversely, every Artinian integral domain is a field. In particular, all finite integral domains are finite fields (more generally, by Wedderburn's little theorem, finite domains are finite fields). The ring of integers Z provides an example of a non-Artinian infinite integral domain that is not a field, possessing infinite descending sequences of ideals such as:

• Rings of polynomials are integral domains if the coefficients come from an integral domain. For instance, the ring Z of all polynomials in one variable with integer coefficients is an integral domain; so is the ring R of all polynomials in two variables with real coefficients.
• For each integer n > 1, the set of all real numbers of the form a + b√n with a and b integers is a subring of R and hence an integral domain.
• For each integer n > 0 the set of all complex numbers of the form a + bi√n with a and b integers is a subring of C and hence an integral domain. In the case n = 1 this integral domain is called the Gaussian integers.
• The p-adic integers.
• If U is a connected open subset of the complex number plane C, then the ring H(U) consisting of all holomorphic functions f : U → C is an integral domain. The same is true for rings of analytic functions on connected open subsets of analytic manifolds.
• If R is a commutative ring and P is an ideal in R, then the factor ring R/P is an integral domain if and only if P is a prime ideal. Also, R is an integral domain if and only if the ideal (0) is a prime ideal.
• A regular local ring is an integral domain. In fact, a regular local ring is a UFD.
• An integrally closed domain such as a Dedekind domain.

The following rings are not integral domains.

• The ring of n×n matrices over any non-trivial ring when n ≥ 2.
• The ring of continuous functions on the unit interval.
• The quotient ring Z/mZ when m is a composite number.
• The commutative ring with a multiplicative identity Z×Z.