Integral domain

In mathematics, and specifically in abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero.^[1]^[2] Integral domains are generalizations of the ring of integers and provide a natural setting for studying divisibility. In an integral domain every nonzero element a has the cancellation property, that is, if a ≠ 0, an equality ab = ac implies b = c.

"Integral domain" is defined almost universally as above, but there is some variation. This article follows the convention that rings have a multiplicative identity, generally denoted 1, but some authors do not follow this, by not requiring integral domains to have a multiplicative identity.^[3]^[4] Noncommutative integral domains are sometimes admitted.^[5] This article, however, follows the much more usual convention of reserving the term "integral domain" for the commutative case and using "domain" for the general case including noncommutative rings.

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 ⊃ integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ finite fields

Algebraic structures
Group-like Semigroup / Monoid Racks and quandles Quasigroup and loop Abelian group Magma Lie group Group theory
Ring-like Ring Semiring Near-ring Commutative ring Integral domain Field Division ring Ring theory
Lattice-like Lattice Semilattice Map of lattices Lattice theory
Module-like Module Group with operators Vector space Linear algebra
Algebra-like Algebra Associative Non-associative Composition algebra Lie algebra Graded Bialgebra

Definitions

There are a number of equivalent definitions of integral domain:

An integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero.
An integral domain is a nonzero commutative ring with no nonzero zero divisors.
An integral domain is a commutative ring in which the zero ideal {0} is a prime ideal.
An integral domain is a nonzero commutative ring for which every non-zero element is cancellable under multiplication.
An integral domain is a ring for which the set of nonzero elements is a commutative monoid under multiplication (because the monoid is closed under multiplication).
An integral domain is a ring that is (isomorphic to) a subring of a field. (This implies it is a nonzero commutative ring.)
An integral domain is a nonzero commutative ring in which for every nonzero element r, the function that maps each element x of the ring to the product xr is injective. Elements r with this property are called regular, so it is equivalent to require that every nonzero element of the ring be regular.

Examples

The archetypical example is the ring $\mathbb {Z}$ of all integers.
Every field is an integral domain. For example, the field $\mathbb {R}$ of all real numbers 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 $\mathbb {Z}$ provides an example of a non-Artinian infinite integral domain that is not a field, possessing infinite descending sequences of ideals such as:

\mathbb {Z} \;\supset \;2\mathbb {Z} \;\supset \;\cdots \;\supset \;2^{n}\mathbb {Z} \;\supset \;2^{n+1}\mathbb {Z} \;\supset \;\cdots

Rings of polynomials are integral domains if the coefficients come from an integral domain. For instance, the ring $\mathbb {Z} [x]$ of all polynomials in one variable with integer coefficients is an integral domain; so is the ring $\mathbb {C} [x_{1},\ldots ,x_{n}]$ of all polynomials in n-variables with complex coefficients.
The previous example can be further exploited by taking quotients from prime ideals. For example, the ring $\mathbb {C} [x,y]/(y^{2}-x(x-1)(x-2))$ corresponding to a plane elliptic curve is an integral domain. Integrality can be checked by showing $y^{2}-x(x-1)(x-2)$ is an irreducible polynomial.
The ring $\mathbb {Z} [x]/(x^{2}-n)\cong \mathbb {Z} [{\sqrt {n}}]$ is an integral domain for any non-square integer $n$ . If $n>0$ then this ring is always a subring of $\mathbb {R}$ , otherwise, it is a subring of $\mathbb {C}$ .
The ring of p-adic integers $\mathbb {Z} _{p}$ is an integral domain.
If $U$ is a connected open subset of the complex plane $\mathbb {C}$ , then the ring ${\mathcal {H}}(U)$ consisting of all holomorphic functions is an integral domain. The same is true for rings of analytic functions on connected open subsets of analytic manifolds.
A regular local ring is an integral domain. In fact, a regular local ring is a UFD.^[7]^[8]

Non-examples

The following rings are not integral domains.

The zero ring (the ring in which $0=1$ ).
The quotient ring Z/mZ when m is a composite number. Indeed, choose a proper factorization $m=xy$ (meaning that $x$ and $y$ are not equal to $1$ or $m$ ). Then $x\not \equiv 0{\pmod {m}}$ and $y\not \equiv 0{\pmod {m}}$ , but $xy\equiv 0{\pmod {m}}$ .
If R and S are commutative rings, then the product ring $R \times S$ is never an integral domain, because $(1,0)\cdot (0,1)=(0,0)$ is the zero element.
When $n$ is a square, the ring $\mathbb {Z} [x]/(x^{2}-n)$ is not an integral domain. Write $n=m^{2}$ , and note that there is a factorization $x^{2}-n=(x-m)(x+m)$ in $\mathbb {Z} [x]$ . By the Chinese remainder theorem, there is an isomorphism $\mathbb {Z} [x]/(x^{2}-n)\cong \mathbb {Z} [x]/(x-m)\times \mathbb {Z} [x]/(x+m)\cong \mathbb {Z} \times \mathbb {Z}$ .
The ring of n × n matrices over any nonzero ring when n ≥ 2. If $M$ and $N$ are matrices such that the image of $N$ is contained in the kernel of $M$ , then $MN=0$ . For example, this happens for $M=N=({\begin{smallmatrix}0&1\\0&0\end{smallmatrix}})$ .
A geometric non-example is the quotient ring $k[x_{1},\ldots ,x_{n}]/(fg)$ for any field $k$ , since $(fg)$ is not a prime ideal. This is geometric because the underlying topological space of the affine scheme $\operatorname {Spec} k[x_{1},\ldots ,x_{n}]/(fg)$ is reducible, since it is the union of the closed subsets $\{f=0\}$ and $\{g=0\}$ .
The ring of continuous functions on the unit interval. Consider the function $f$ which is $1-2x$ on $[0,1/2]$ and zero on $[1/2,1]$ and the function $g$ which is zero on $[0,1/2]$ and $2x-1$ on $[1/2,1]$ . Neither $f$ nor $g$ is everywhere zero, but $fg$ is.
The tensor product $\mathbb {C} \otimes _{\mathbb {R} }\mathbb {C}$ . This ring has two non-trivial idempotents, $\textstyle e_{1}={\frac {1}{2}}(1\otimes 1)-{\frac {1}{2}}(i\otimes i)$ and $\textstyle e_{2}={\frac {1}{2}}(1\otimes 1)+{\frac {1}{2}}(i\otimes i)$ . They are orthogonal, meaning that $e_{1}e_{2}=0$ , and hence $\mathbb {C} \otimes _{\mathbb {R} }\mathbb {C}$ is not a domain. In fact, there is an isomorphism $\mathbb {C} \times \mathbb {C} \to \mathbb {C} \otimes _{\mathbb {R} }\mathbb {C}$ defined by $(z,w)\mapsto z\cdot e_{1}+w\cdot e_{2}$ . Its inverse is defined by $z\otimes w\mapsto (zw,z{\bar {w}})$ . This example shows that a fiber product of irreducible affine schemes need not be irreducible.

Divisibility, prime elements, and irreducible elements

In this section, R is an integral domain.

Given elements a and b of R, we say that a divides b, or that a is a divisor of b, or that b is a multiple of a, if there exists an element x in R such that ax = b.

The elements that 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.^[9] Equivalently, a and b are associates if a=ub for some unit u.

If q is a nonzero 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 nonzero 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. Equivalently, an element p is prime if and only if the principal ideal (p) is a nonzero prime ideal. The notion of prime element generalizes the ordinary definition of prime number in the ring Z, except that it allows for negative prime elements.

Every prime element is irreducible. The converse is not true in general: for example, in the quadratic integer ring $\mathbb {Z} \left[{\sqrt {-5}}\right]$ the element 3 is irreducible (if it factored nontrivially, the factors would each have to have norm 3, but there are no norm 3 elements since $a^{2}+5b^{2}=3$ has no integer solutions), but not prime (since 3 divides $\left(2+{\sqrt {-5}}\right)\left(2-{\sqrt {-5}}\right)$ without dividing either factor). In a unique factorization domain (or more generally, a GCD domain), an irreducible element is a prime element.

While unique factorization does not hold in $\mathbb {Z} \left[{\sqrt {-5}}\right]$ , there is unique factorization of ideals. See Lasker–Noether theorem.

Properties

A commutative ring R is an integral domain if and only if the ideal (0) of R is a prime ideal. This implies the associated affine scheme is reduced and irreducible. If $R$ is graded then the associated projective scheme is also reduced and irreducible.
If R is a commutative ring and P is an ideal in R, then the quotient ring R/P is an integral domain if and only if P is a prime ideal.
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 any integral domain: for any a, b, and c in 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 nonzero a in the domain.
The cancellation property holds for ideals in any integral domain: if xI = xJ, then either x is zero or I = J.
An integral domain is equal to the intersection of its localizations at maximal ideals.
An inductive limit of integral domains is an integral domain.

Field of fractions

The field of fractions K of an integral domain R is the set of fractions a/b with a and b in R and b ≠ 0 modulo an appropriate equivalence relation, equipped with the usual addition and multiplication operations. It is "the smallest field containing R" in the sense that there is an injective ring homomorphism R → K such that any injective ring homomorphism from R to a field factors through K. The field of fractions of the ring of integers Z is the field of rational numbers Q. The field of fractions of a field is isomorphic to the field itself.

Algebraic geometry

Integral domains are characterized by the condition that they are reduced (that is x² = 0 implies x = 0) and irreducible (that is there is only one minimal prime ideal). 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, so such rings are integral domains. The converse is clear: an integral domain has no nonzero nilpotent elements, and the zero ideal is the unique minimal prime ideal.

This translates, in algebraic geometry, into the fact that the coordinate ring of an affine algebraic set is an integral domain if and only if the algebraic set is an algebraic variety.

More generally, a commutative ring is an integral domain if and only if its spectrum is an integral affine scheme.

Characteristic and homomorphisms

The characteristic of an integral domain is either 0 or a prime number.

If R is an integral domain of prime characteristic p, then the Frobenius endomorphism f(x) = x^p is injective.

Notes

↑ Bourbaki, p. 116.
↑ Dummit and Foote, p. 228.
↑ B.L. van der Waerden, Algebra Erster Teil, p. 36, Springer-Verlag, Berlin, Heidelberg 1966.
↑ I.N. Herstein, Topics in Algebra, p. 88-90, Blaisdell Publishing Company, London 1964.
↑ J.C. McConnel and J.C. Robson "Noncommutative Noetherian Rings" (Graduate Studies in Mathematics Vol. 30, AMS)
↑ Pages 91–92 of Lang, Serge (1993), Algebra (Third ed.), Reading, Mass.: Addison-Wesley, ISBN 978-0-201-55540-0, Zbl 0848.13001
↑ Auslander, Maurice; Buchsbaum, D. A. (1959). "Unique factorization in regular local rings". Proc. Natl. Acad. Sci. USA. 45 (5): 733–734. PMC 222624 . PMID 16590434. doi:10.1073/pnas.45.5.733.
↑ Masayoshi Nagata (1958). "A general theory of algebraic geometry over Dedekind domains. II". Amer. J. Math. The Johns Hopkins University Press. 80 (2): 382–420. JSTOR 2372791. doi:10.2307/2372791.
↑ Durbin, John R. (1993). Modern Algebra: An Introduction (3rd ed.). John Wiley and Sons. p. 224. ISBN 0-471-51001-7. Elements a and b of [an integral domain] are called associates if a | b and b | a.

References

Adamson, Iain T. (1972). Elementary rings and modules. University Mathematical Texts. Oliver and Boyd. ISBN 0-05-002192-3.
Bourbaki, Nicolas (1998). Algebra, Chapters 1–3. Berlin, New York: Springer-Verlag. ISBN 978-3-540-64243-5.
Mac Lane, Saunders; Birkhoff, Garrett (1967). Algebra. New York: The Macmillan Co. ISBN 1-56881-068-7. MR 0214415.
Dummit, David S.; Foote, Richard M. (2004). Abstract Algebra (3rd ed.). New York: Wiley. ISBN 978-0-471-43334-7.
Hungerford, Thomas W. (2013). Abstract Algebra: An Introduction (3rd ed.). Cengage Learning. ISBN 978-1-111-56962-4.
Lang, Serge (2002). Algebra. Graduate Texts in Mathematics. 211. Berlin, New York: Springer-Verlag. ISBN 978-0-387-95385-4. MR 1878556.
Sharpe, David (1987). Rings and factorization. Cambridge University Press. ISBN 0-521-33718-6.
Rowen, Louis Halle (1994). Algebra: groups, rings, and fields. A K Peters. ISBN 1-56881-028-8.
Lanski, Charles (2005). Concepts in abstract algebra. AMS Bookstore. ISBN 0-534-42323-X.
Milies, César Polcino; Sehgal, Sudarshan K. (2002). An introduction to group rings. Springer. ISBN 1-4020-0238-6.
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