Integral domain

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In abstract algebra, an integral domain is a commutative ring with 0 ≠ 1 in which the product of any two non-zero elements is always non-zero; that is, there are no zero divisors. Integral domains are generalizations of the integers and provide a natural setting for studying divisibility.

Alternatively and equivalently, integral domains may be defined as commutative rings in which the zero ideal {0} is prime, or as the subrings of fields. Viewing the underlying commutative ring as a categorical construction, the previous criterion on zero divisors is equivalent to the condition that every nonzero morphism is a monomorphism (hence also an epimorphism).

The condition 0 ≠ 1 only serves to exclude the trivial ring {0} with a single element.

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, the only finite integral domains are the finite fields.

Rings of polynomials are integral domains if the coefficients come from an integral domain. For instance, the ring Z[X] of all polynomials in one variable with integer coefficients is an integral domain; so is the ring R[X,Y] of all polynomials in two variables with real coefficients.

The set of all real numbers of the form a + b√2 with a and b integers is a subring of R and hence an integral domain. A similar example is given by the complex numbers of the form a + bi with a and b integers (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 analytical functions on connected open subsets of analytical 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.

Divisibility, prime and irreducible elements

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.

If a divides b and b divides c, then a divides c. If a divides b, then a divides every multiple of b. If a divides two elements, then a also divides their sum and difference.

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. a and b are associated if and only if there exists a unit u such that au = b.

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 b.

This generalizes the ordinary definition of prime number in the ring Z, except that it allows for negative prime elements. If p is a prime element, then the principal ideal (p) generated by p is a prime ideal. Every prime element is irreducible (here, for the first time, we need R to be an integral domain), but the converse is not true in all integral domains (it is true in unique factorization domains, however).

Field of fractions

If R is a given integral domain, the smallest field Quot(R) containing R as a subring is uniquely determined up to isomorphism and is called the field of fractions or quotient field of R. It consists 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.

Characteristic and homomorphisms

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 homomorphism.

See also

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