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. 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.
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
- Integral domains - wikibook link
Category:Commutative algebra
Category:Ring theory
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