In abstract algebra, an integral domain is a commutative ring that has no zero divisors,[1] and which is not the trivial ring {0}. It is usually assumed that commutative rings and integral domains have a multiplicative identity even though this is not always included in the definition of a ring. Integral domains are generalizations of the integers and provide a natural setting for studying divisibility. An integral domain is a commutative domain with identity.[2]
The above is how "integral domain" is almost universally defined, but there is some variation. In particular, noncommutative integral domains are sometimes admitted.[3] However, we follow the much more usual convention of reserving the term integral domain for the commutative case and use domain for the noncommutative case; this implies that, curiously, the adjective "integral" means "commutative" in this context. Some sources, notably Lang, use the term entire ring for integral domain.[4]
Some specific kinds of integral domains are given with the following chain of class inclusions:
The absence of zero divisors means that in an integral domain the cancellation property holds for multiplication by any nonzero element a: an equality ab = ac implies b = c.
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There are a number of equivalent definitions of integral domain:
The following rings are not integral domains.
If a and b are elements of the integral domain R, we say that a divides b or a is a divisor of b or b is a multiple of a if and only if there exists an element x in R such that ax = b.
The elements which divide 1 are called the units of R; these are precisely the invertible elements in R. Units divide all other elements.
If a divides b and b divides a, then we say a and b are associated elements or associates.
If q is a non-unit, we say that q is an irreducible element if q cannot be written as a product of two non-units.
If p is a non-zero non-unit, we say that p is a prime element if, whenever p divides a product ab, then p divides a or p divides b. Equivalent, an element is prime if and only if an ideal generated by it is a nonzero prime ideal. Every prime element is irreducible. Conversely, in a GCD domain (e.g., a unique factorization domain), an irreducible element is a prime element.
The notion of prime element generalizes the ordinary definition of prime number in the ring Z, except that it allows for negative prime elements. While every prime is irreducible, the converse is not in general true. For example, in the quadratic integer ring the number 3 is irreducible, but is not a prime because 9, the norm of 3, can be factored in two ways in the ring, namely, and . Thus , but 3 does not divide nor The numbers 3 and are irreducible as there is no where or as has no integer solution.
While unique factorization does not hold in the above example, if we use ideals we do get unique factorization. Namely, the ideal equals the ideals and and is the unique product of the two prime ideals: , each of which have a norm of 3.
If R is a given integral domain, the smallest field containing R as a subring is uniquely determined up to isomorphism and is called the field of fractions or quotient field of R. It can be thought of as consisting of all fractions a/b with a and b in R and b ≠ 0, modulo an appropriate equivalence relation. The field of fractions of the integers is the field of rational numbers. The field of fractions of a field is isomorphic to the field itself.
In algebraic geometry, integral domains correspond to irreducible varieties. They have a unique generic point, given by the zero ideal. Integral domains are also characterized by the condition that they are reduced and irreducible. The former condition ensures that the nilradical of the ring is zero, so that the intersection of all the ring's minimal primes is zero. The latter condition is that the ring have only one minimal prime. It follows that the unique minimal prime ideal of a reduced and irreducible ring is the zero ideal, hence such rings are integral domains. The converse is clear: No integral domain can have nilpotent elements, and the zero ideal is the unique minimal prime ideal.
The characteristic of every integral domain is either zero or a prime number.
If R is an integral domain with prime characteristic p, then f(x) = x p defines an injective ring homomorphism f : R → R, the Frobenius endomorphism.