In abstract algebra, an integral domain is a commutative ring without zero divisors and with a multiplicative identity 1 not equal to 0, the additive identity.[1][2][3][4][5] Integral domains are generalizations of the integers and provide a natural setting for studying divisibility. An integral domain is a commutative domain.
Alternatively and equivalently, an integral domain may be defined as a commutative ring (with unit) in which the zero ideal {0} is prime, or as a subring of a field. Additionally, a commutative ring with unit R is an integral domain if and only if for every non-zero element r of the ring, the R-module map induced by multiplication by r is injective (such r are called regular).
Viewing the underlying commutative ring as a preadditive category, the above criterion on zero divisors is equivalent to the condition that every nonzero morphism is a monomorphism (hence also an epimorphism, by making use of the bilinear structure on the set of morphisms).
The condition 0 ≠ 1 only serves to exclude the trivial ring {0}.
A few sources talk about noncommutative integral domains, but we reserve the term integral domain for the commutative case and use domain for the noncommutative case. Some sources, notably Lang, use the term entire ring for integral domain[6]
Some specific kinds of integral domains are given with the following chain of class inclusions:
- Commutative rings ⊃ integral domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields
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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[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.
- 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 is a prime ideal.
- A regular local ring is an integral domain. A deep theorem of Auslander-Buchsbaum formula[7] and Nagata[8] from the 1950s claims that, in fact, a regular local ring is a UFD.
Counter-examples
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/m when m is a composite number.
- The commutative ring with a multiplicative identity Z×Z.
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 or associates. 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 p divides b. Equivalent, an element is prime if and only if an ideal generated by it is a nonzero prime ideal.
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). For example, in the quadratic integer ring the number 3 is irreducible but is not a prime because 9 can be written as and 3 * 3.
Being prime is also relative to which ring an element is considered to be in; for example, 2 is a prime element in Z but it is not in Z[i], the ring of Gaussian integers, since 2 = (1 + i)(1 − i).
Properties
- Let R be an integral domain. Then there is an integral domain S such that R ⊂ S and S has an element which is transcendental over R.
- The cancellation property holds in integral domains. That is, let a, b, and c belong to an integral domain. If a ≠ 0 and ab = ac then b = c. Another way to state this is that the function x ax is injective for any non-zero a in the domain. (Recall from vector algebra that a transformation T is injective if and only if its null space consists of 0 alone. It is therefore possible to have a ring-isomorphic module with non-injective T — if the ring is not an integral domain.)
- An integral domain is equal to the intersection of its localizations at maximal ideals.
Field of fractions
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.
Algebraic geometry
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 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 ideal.
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 endomorphism.
See also
- Integral domains - wikibook link
- Dedekind–Hasse norm - the extra structure needed for an integral domain to be principal
- Zero-product property
Notes
- ^ Dummit and Foote, p. 229
- ^ Hungerford, p. 116
- ^ Lang, p. 91
- ^ Birkhoff and Mac Lane, p. 133
- ^ Bourbaki, p. 116
- ^ Pages 91-92 of Lang, Serge (1993), Algebra (Third ed.), Reading, Mass.: Addison-Wesley Pub. Co., ISBN 978-0-201-55540-0
- ^ Maurice Auslander; D.A. Buchsbaum (1959). "Unique factorization in regular local rings". Proc. Natl. Acad. Sci. USA 45: 733–734. doi:10.1073/pnas.45.5.733. PMID 16590434.
- ^ Masayoshi Nagata (1958). "A general theory of algebraic geometry over Dedekind domains. II". Amer. J. Math. 80: 382–420. doi:10.2307/2372791.
References
- Iain T. Adamson (1972). Elementary rings and modules. University Mathematical Texts. Oliver and Boyd. ISBN 0-05-002192-3.
- Bourbaki, Nicolas (1988), Algebra, Berlin, New York: Springer-Verlag, ISBN 978-3-540-19373-9
- Mac Lane, Saunders; Birkhoff, Garrett (1967), Algebra, New York: The Macmillan Co., MR 0214415
- Dummit, David S.; Foote, Richard M. (1999), Abstract algebra (2nd ed.), New York: John Wiley & Sons, ISBN 978-0-471-36857-1
- Hungerford, Thomas W. (1974), Algebra, New York: Holt, Rinehart and Winston, Inc.
- Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, 211, Berlin, New York: Springer-Verlag, MR 1878556, ISBN 978-0-387-95385-4
- David Sharpe (1987). Rings and factorization. Cambridge University Press. ISBN 0-521-33718-6.
Categories: Commutative algebra | Ring theory
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