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Integral domain

An integral domain is a nonzero commutative ring with a multiplicative identity element such that the product of any two nonzero elements is nonzero, meaning it has no zero divisors.^[1] This structure generalizes the familiar ring of integers \mathbb{Z}, where multiplication behaves without "accidental" zeros, ensuring a form of cancellation property: if ab = ac and a \neq 0, then b = c.^[2] Common examples of integral domains include the integers \mathbb{Z}, the rational numbers \mathbb{Q}, and polynomial rings k over a field k, such as \mathbb{R}.^[3] Fields like \mathbb{Q} and \mathbb{R} are special cases of integral domains, where every nonzero element has a multiplicative inverse. Non-examples include rings like \mathbb{Z}/6\mathbb{Z}, which have zero divisors such as 2 and 3, since $2 \cdot 3 = 0.^[4] Integral domains form the foundational building blocks in commutative algebra, enabling the study of ideals, prime elements, and factorization properties.^[5] They support constructions like the field of fractions, which embeds the domain into a field by adjoining inverses for nonzero elements.^[6] Advanced subclasses, such as principal ideal domains (PIDs) and unique factorization domains (UFDs), extend these properties to guarantee unique factorizations up to units, playing crucial roles in algebraic number theory and geometry.^[7]

Fundamentals

Definition

An integral domain is defined as a commutative ring R with a multiplicative identity $1 \neq [0](/page/0) such that there are no zero divisors; that is, for all a, b \in R, if ab = [0](/page/0), then either a = [0](/page/0) or b = [0](/page/0).^[8] This condition ensures that the ring behaves in a manner analogous to the integers under multiplication, avoiding the complications introduced by zero divisors in more general rings.^[9] The commutativity requirement specifies that multiplication in R satisfies ab = ba for all a, b \in R, while the unity element $1 acts as the multiplicative identity, satisfying $1 \cdot a = a \cdot 1 = a for every a \in R. The stipulation that $1 \neq 0 excludes the trivial zero ring from consideration. A key consequence of the absence of zero divisors is the cancellation law: if ab = ac and a \neq 0, then b = c. This follows directly from the no-zero-divisors property, as a(b - c) = 0 implies b - c = 0.^[8] The term "integral domain" originates from the German "Integritätsbereich," introduced in the context of algebraic number theory by David Hilbert around 1900, reflecting the integrity of the ring structure without "cracks" caused by zero divisors.^[10] Integral domains are commonly denoted simply as such or as commutative rings without zero divisors to emphasize their defining characteristic.

Examples

The ring of integers \mathbb{[Z](/page/Z)} serves as the prototypical example of an integral domain, where multiplication of nonzero elements always yields a nonzero product, ensuring the absence of zero divisors.^[11] This structure underpins much of elementary number theory, as any two nonzero integers a and b satisfy ab \neq 0.^[12] Polynomial rings over fields provide another fundamental class of integral domains; for a field k, the ring k consists of polynomials in one indeterminate x with coefficients in k, and it has no zero divisors because the degree of a product of nonzero polynomials equals the sum of their degrees, which is positive.^[13] This property holds similarly for multivariate polynomial rings k[x_1, \dots, x_n] over a field k.^[11] The Gaussian integers \mathbb{Z} = \{a + bi \mid a, b \in \mathbb{Z}\}, which are complex numbers with integer real and imaginary parts under the usual addition and multiplication, form an integral domain, as the product of two nonzero elements is nonzero, inheriting this from the field of complex numbers.^[14] This ring extends the integers to the quadratic field \mathbb{Q}(i) and exhibits unique factorization into Gaussian primes.^[15] More generally, the ring of algebraic integers in a number field K, denoted \mathcal{O}_K, comprises all elements of K that are roots of monic polynomials with integer coefficients and forms an integral domain as a subring of the field K.^[16] For instance, in quadratic fields \mathbb{Q}(\sqrt{d}) for square-free integer d, \mathcal{O}_K includes rings like \mathbb{Z}[\sqrt{d}] or \mathbb{Z}\left[\frac{1 + \sqrt{d}}{2}\right] depending on d modulo 4, each without zero divisors.^[17] In algebraic geometry, the coordinate ring of an irreducible affine variety over an algebraically closed field k is the quotient of a polynomial ring k[x_1, \dots, x_n] by a prime ideal, rendering it an integral domain that encodes the polynomial functions on the variety.^[18] This construction ties commutative algebra to geometry, where irreducibility ensures no zero divisors in the ring.^[19]

Non-examples

While integral domains are commutative rings with multiplicative identity and no zero divisors, several familiar ring structures fail one or both of these conditions, serving as non-examples. Matrix rings over fields provide a prominent illustration. For n \geq 2, the ring M_n(k) of n \times n matrices over a field k is a non-commutative ring with unity, but it contains zero divisors; for instance, the product of two non-zero matrices can be the zero matrix, as seen with matrix units E_{11} and E_{22} (with 1's on the respective diagonal entries and zeros elsewhere), where E_{11} E_{22} = 0 but neither is zero.^[20] Quotient rings of the integers by non-prime ideals also fail the no-zero-divisors property. The ring \mathbb{Z}/4\mathbb{Z} is a commutative ring with unity, but it has zero divisors: the element $2 + 4\mathbb{Z} satisfies (2 + 4\mathbb{Z})^2 = 0 + 4\mathbb{Z}, yet $2 + 4\mathbb{Z} \neq 0 + 4\mathbb{Z}.^[21] Direct products of integral domains similarly introduce zero divisors despite retaining commutativity and unity. The ring \mathbb{Z} \times \mathbb{Z}, with componentwise addition and multiplication, is commutative with unity (1,1), but (1,0) \cdot (0,1) = (0,0), where neither factor is zero.^[22] Finally, rings lacking a multiplicative identity cannot qualify as integral domains, even if they are commutative and free of zero divisors. The even integers $2\mathbb{Z} under standard addition and multiplication form such a structure, with no element serving as a unity since for any $2k \in 2\mathbb{Z}, $2k \cdot 1 = 2k \neq 1.^[23]

Divisibility and Factorization

Prime Elements

In an integral domain R, a nonzero non-unit element p \in R is called a prime element if whenever p divides the product ab for a, b \in R, then p divides a or p divides b.^[24] This property ensures that prime elements behave analogously to prime numbers in the integers, capturing a strong form of divisibility that is essential for studying factorization within the domain.^[25] The principal ideal generated by a prime element p, denoted (p), is a prime ideal in R. Conversely, if (p) is a nonzero prime ideal that is principal, then p is a prime element.^[26] This ideal-theoretic characterization links the algebraic structure of prime elements to the broader theory of ideals in commutative rings, highlighting their role in the spectrum of the domain.^[26] A fundamental theorem states that in any integral domain, every prime element generates a prime ideal, reinforcing the equivalence between the element-wise and ideal-based definitions.^[26] For example, in the ring of integers \mathbb{Z}, the prime elements are precisely the prime numbers such as 2, 3, and 5, each generating a prime ideal like (2), which consists of all even integers.^[24] While every prime element in an integral domain is irreducible (meaning it cannot be factored into non-unit elements), the converse does not hold in general domains, setting the stage for further distinctions in factorization theory.^[27]

Irreducible Elements

In an integral domain R, a non-zero non-unit element r \in R is called irreducible if whenever r = ab for some a, b \in R, then either a or b is a unit in R.^[28] This condition captures the notion that r cannot be factored non-trivially into non-units, making it a basic building block within the ring's multiplicative structure.^[28] Within factorization theory in commutative algebra, irreducible elements are synonymous with atoms, serving as the indivisible components from which non-unit elements can be expressed as finite products, provided the domain admits such decompositions.^[29] This atomic perspective underpins much of the study of unique and non-unique factorizations in integral domains, emphasizing the role of irreducibles in constructing longer factor chains.^[29] Prime elements, which satisfy a stronger divisibility condition, form a subset of the irreducible elements.^[30] Representative examples illustrate these concepts clearly. In the ring of integers \mathbb{Z}, the irreducible elements coincide with the positive prime numbers (considering units \pm 1), such as 2, 3, and 5, as any factorization of a prime must involve a unit.^[28] Similarly, in the polynomial ring k[x, y] over a field k, the element x is irreducible, since any factorization x = f g with f, g \in k[x, y] requires one of f or g to be a unit (a non-zero constant in k).^[31] In principal ideal domains, the principal ideal generated by an irreducible element is primary, as it coincides with a prime ideal in such rings.^[28]

Units and Associates

In an integral domain R, a unit is a nonzero element u \in R that has a multiplicative inverse in R, meaning there exists v \in R such that uv = 1.^[7] The set of all units in R, denoted U(R), forms an abelian group under multiplication, with the identity element serving as the multiplicative identity of R.^[32] Two nonzero elements a, b \in R are associates, written a \sim b, if there exists a unit u \in U(R) such that a = ub.^[7] This relation is an equivalence relation on the nonzero elements of R: it is reflexive (taking u = 1), symmetric (since if u is a unit, so is u^{-1}), and transitive (composing units yields another unit).^[32] Equivalently, a \sim b if and only if each divides the other, i.e., a \mid b and b \mid a.^[32] Associates in an integral domain share several key properties related to divisibility and ideals. In particular, the principal ideals generated by associates are equal: (a) = (b) if and only if a \sim b.^[7] Moreover, in the context of factorization, irreducible elements are considered up to associates, meaning that factorizations into irreducibles are unique modulo multiplication by units and reordering.^[7] A concrete example occurs in the ring of integers \mathbb{Z}, where the units are \pm 1, and thus n \sim -n for any nonzero integer n, such as $2 \sim -2.^[7] In the Gaussian integers \mathbb{Z}, the units are $1, -1, i, -i, so elements like $1 + i and i(1 + i) = i - 1 are associates.^[32] The associate relation partitions the nonzero elements of R into equivalence classes determined precisely by the unit group U(R): two elements lie in the same class if and only if their ratio is a unit.^[32] This structure underscores the role of units in classifying elements up to multiplication in integral domains.^[7]

Properties and Classifications

General Properties

Integral domains exhibit several fundamental structural properties that distinguish them from more general commutative rings. A key feature is the cancellation law: in an integral domain D, for any nonzero a \in D and any b, c \in D, if ab = ac, then b = c.^[33] This follows directly from the absence of zero divisors, as a(b - c) = 0 implies b - c = 0 since a \neq 0. Additionally, subrings of integral domains inherit this property; if R is a subring of an integral domain S, then R is itself an integral domain.^[3] Similarly, quotients by prime ideals preserve the integral domain structure: if P is a prime ideal of an integral domain R, then the quotient ring R/P is an integral domain.^[34] Another important class within integral domains are those that are integrally closed. An integral domain R with field of fractions K is integrally closed if every element of K that is integral over R—meaning it satisfies a monic polynomial with coefficients in R—actually belongs to R.^[35] The ring of integers \mathbb{Z} provides a classic example, as it is integrally closed in \mathbb{Q}; any algebraic integer in \mathbb{Q} must be an ordinary integer.^[35] Regarding chain conditions on ideals, integral domains interact distinctly with Noetherian and Artinian properties. A Noetherian integral domain satisfies the ascending chain condition on ideals, ensuring that every ascending sequence of ideals stabilizes. However, an Artinian integral domain—satisfying the descending chain condition—must be a field; otherwise, for a nonzero nonunit x \in R, the chain (x) \supseteq (x^2) \supseteq (x^3) \supseteq \cdots descends infinitely without stabilization, contradicting Artinianity.^[36] Finally, every integral domain admits a field of fractions, a field containing the domain as a subring where every nonzero element becomes invertible. This universal property embeds the domain into a field, providing a canonical extension for studying fractions and localizations.^[37]

Characteristic

The characteristic of an integral domain R is defined as the smallest positive integer n such that n \cdot 1 = 0 in R, where $1 denotes the multiplicative identity element of R; if no such positive integer exists, the characteristic is $0.^[11] This notion arises from the canonical ring homomorphism \mathbb{Z} \to R sending k \mapsto k \cdot 1, whose kernel is the principal ideal generated by the characteristic.^[3] In an integral domain, the characteristic must be either $0 or a prime number p. Suppose the characteristic n > 1 is composite, so n = ab with integers a, b > 1. Then a \cdot 1 \neq 0 and b \cdot 1 \neq 0 (as n is minimal), but (a \cdot 1)(b \cdot 1) = n \cdot 1 = 0, contradicting the absence of zero divisors in R. Thus, no such composite n can occur, leaving only prime characteristics or $0.^[15] Examples illustrate this property clearly. The ring of integers \mathbb{Z} has characteristic $0, as multiples of $1 never yield $0. For the polynomial ring k over a field k, the characteristic equals that of k, since the homomorphism \mathbb{Z} \to k factors through k and inherits the same kernel.^[11]^[38] Ring homomorphisms between integral domains preserve the characteristic. Specifically, for a unital homomorphism \phi: R \to S, we have \phi(n \cdot 1_R) = n \cdot 1_S, so if n \cdot 1_R = 0 then n \cdot 1_S = 0; moreover, in the injective case (such as subdomain inclusions), the minimality ensures equality of characteristics.^[39] A key structural implication arises in positive characteristic: if R has prime characteristic p, then R admits the structure of a vector space over the prime field \mathbb{F}_p. The embedding \mathbb{F}_p \to R given by $1 \mapsto 1_R is injective (as the kernel would contradict the characteristic being exactly p), allowing scalar multiplication by elements of \mathbb{F}_p via repeated addition in R.^[40]

Types of Integral Domains

Integral domains can be classified into several important subclasses based on their ideal structure and factorization properties. These classifications highlight domains with enhanced properties that facilitate algebraic manipulations, such as unique factorization or the ability to generate all ideals from single elements. Key types include principal ideal domains, unique factorization domains, Euclidean domains, and Bézout domains, each building upon the basic structure of an integral domain while imposing additional conditions that lead to stronger theorems and applications in algebra./18%3A_Integral_Domains/18.02%3A_Factorization_in_Integral_Domains) A principal ideal domain (PID) is an integral domain in which every ideal is principal, meaning it can be generated by a single element./02%3A_Fields_and_Rings/2.04%3A_Principal_Ideals_and_Euclidean_Domains) Classic examples include the ring of integers \mathbb{Z}, where every ideal is of the form (n) for some n \in \mathbb{Z}, and the polynomial ring k over a field k, where ideals are generated by single polynomials./02%3A_Fields_and_Rings/2.04%3A_Principal_Ideals_and_Euclidean_Domains) In PIDs, the structure simplifies many proofs, such as those involving greatest common divisors, because any two elements generate an ideal that is principal./18%3A_Integral_Domains/18.02%3A_Factorization_in_Integral_Domains) A unique factorization domain (UFD) is an integral domain where every nonzero non-unit element can be factored into a product of irreducible elements, and this factorization is unique up to the order of factors and associates (elements differing by multiplication by units)./03%3A_Factorization/3.02%3A_Factorization_in_Euclidean_Domains) The integers \mathbb{Z} and polynomial rings k over fields are UFDs, but not all UFDs are PIDs; for instance, the polynomial ring \mathbb{Z} is a UFD because it inherits unique factorization from \mathbb{Z} via Gauss's lemma, yet it contains non-principal ideals like (2, x)./18%3A_Integral_Domains/18.02%3A_Factorization_in_Integral_Domains) This property ensures that irreducibles behave like primes in factorization, enabling reliable decomposition in algebraic number theory./03%3A_Factorization/3.02%3A_Factorization_in_Euclidean_Domains) An Euclidean domain is an integral domain equipped with a Euclidean function (a norm mapping to non-negative integers) that allows a division algorithm: for any a, b \neq 0, there exist q, r such that a = qb + r with either r = 0 or the norm of r is less than the norm of b./08%3A_Rings_II/8.03%3A_Euclidean_Domains) Examples include \mathbb{Z} with the absolute value norm and k with the degree function, both of which support the Euclidean algorithm for computing gcds./02%3A_Fields_and_Rings/2.04%3A_Principal_Ideals_and_Euclidean_Domains) Euclidean domains are particularly useful for constructive proofs, as the norm enables well-ordering arguments similar to those in natural numbers./08%3A_Rings_II/8.03%3A_Euclidean_Domains) A Bézout domain is an integral domain in which every finitely generated ideal is principal.^[41] This generalizes PIDs by relaxing the condition to finitely generated ideals only, allowing for non-Noetherian examples like the ring of all algebraic integers, where every ideal generated by two elements is principal but not all ideals are.^[41] Another example is the ring of entire functions on the complex plane, which satisfies Bézout's condition due to the ability to solve linear combinations for gcds in such settings.^[42] Bézout domains bridge factorization properties with ideal theory, often appearing in valuation rings or Prüfer domains with additional gcd conditions.^[43] These types form an implication chain: every Euclidean domain is a PID, every PID is a UFD, and every UFD is an integral domain, but the converses do not hold.^[44] For instance, \mathbb{Z}[\sqrt{-5}] is an integral domain but not a UFD, \mathbb{Z} is a UFD but not a PID, and \mathbb{Z}[\frac{1+\sqrt{-19}}{2}] is a PID but not Euclidean.^[44] Bézout domains fit alongside PIDs in the hierarchy, as every PID is Bézout, but non-Noetherian Bézout domains like the algebraic integers exceed PIDs.^[41] This hierarchy underscores how stronger structural assumptions yield more powerful algebraic tools./03%3A_Factorization/3.02%3A_Factorization_in_Euclidean_Domains)

Constructions

Field of Fractions

Every integral domain R admits a field extension known as its field of fractions, denoted \operatorname{Frac}(R), which is constructed as the quotient of the set R \times (R \setminus \{0\}) by the equivalence relation \sim, where (a, b) \sim (c, d) if and only if ad = bc.^[37] This equivalence relation is well-defined precisely because R is an integral domain, ensuring that nonzero elements do not annihilate each other. Elements of \operatorname{Frac}(R) are thus equivalence classes denoted a/b or [a, b], with addition and multiplication defined by [a, b] + [c, d] = [ad + bc, bd] and [a, b] \cdot [c, d] = [ac, bd], respectively; these operations make \operatorname{Frac}(R) into a commutative ring with unity.^[45] The ring \operatorname{Frac}(R) is in fact a field, as every nonzero element [a, b] (with a \neq 0) has a multiplicative inverse [b, a], and the construction yields a field if and only if R is an integral domain.^[46] There is a natural embedding i: R \to \operatorname{Frac}(R) given by i(r) = [r, 1], which is injective because R has no zero divisors, allowing cancellation in equations like r/1 = 0/1 implying r = 0.^[47] In \operatorname{Frac}(R), every nonzero element of the image i(R) becomes invertible, extending the domain to a field while preserving the ring structure of R. The field of fractions satisfies a universal property: for any ring homomorphism \phi: R \to F into a field F, there exists a unique field homomorphism \psi: \operatorname{Frac}(R) \to F such that \psi \circ i = \phi, making \operatorname{Frac}(R) the "universal" field containing R.^[48] This property characterizes \operatorname{Frac}(R) up to unique isomorphism and ensures it is the smallest field extension of R.^[49] Classic examples include the rational numbers \mathbb{Q} = \operatorname{Frac}(\mathbb{Z}), where fractions a/b with a, b \in \mathbb{Z}, b \neq 0, and reduced form represent equivalence classes under the usual rule.^[50] Another is the field of rational functions k(x) = \operatorname{Frac}(k) over a field k, consisting of quotients of polynomials f(x)/g(x) with g(x) \neq 0.^[51] These constructions embed the respective domains faithfully into their fraction fields, illustrating how integral domains can be "completed" to fields.

Homomorphisms

A ring homomorphism \phi: R \to S between integral domains R and S is a map preserving addition, multiplication, and the multiplicative identity, i.e., \phi(r_1 + r_2) = \phi(r_1) + \phi(r_2), \phi(r_1 r_2) = \phi(r_1) \phi(r_2), and \phi(1_R) = 1_S for all r_1, r_2 \in R. The image \phi(R) forms a subring of S, and since S has no zero divisors, any nonzero elements in \phi(R) cannot multiply to zero, making \phi(R) an integral domain. By the first isomorphism theorem, R / \ker(\phi) \cong \phi(R), so \ker(\phi) is a prime ideal of R because the quotient is an integral domain. In particular, if \phi is injective, then \ker(\phi) = \{0\}, the zero ideal, which is prime in any integral domain, ensuring the homomorphism embeds R as a domain inside S. Examples illustrate these properties. The natural inclusion \iota: \mathbb{Z} \to \mathbb{C}, defined by \iota(n) = n \cdot 1_{\mathbb{C}} for n \in \mathbb{Z}, is an injective ring homomorphism with \ker(\iota) = \{0\} and image \mathbb{Z}, both integral domains. Another example is the evaluation homomorphism \mathrm{ev}_a: k \to k for a field k and a \in k, given by \mathrm{ev}_a(f) = f(a), which has kernel the prime ideal (x - a) and image k, an integral domain. A key compatibility result concerns characteristics: for a ring homomorphism \phi: R \to S between integral domains, the characteristic of the image \phi(R) divides the characteristic of R. This holds because \phi induces a homomorphism on the prime subrings \mathbb{Z}/\mathrm{char}(R)\mathbb{Z} \to \mathbb{Z}/\mathrm{char}(S)\mathbb{Z}, and the order of the identity in the additive group of R maps to that of S, implying divisibility when finite.

Applications

Algebraic Geometry

In algebraic geometry, the coordinate ring of an affine variety over an algebraically closed field k plays a central role in bridging algebra and geometry. For an irreducible affine variety V \subset \mathbb{A}^n_k, the coordinate ring k[V] = k[x_1, \dots, x_n]/I(V) is an integral domain, where I(V) is the ideal of polynomials vanishing on V. This equivalence holds because V is irreducible if and only if I(V) is a prime ideal, making the quotient ring free of zero divisors. Conversely, any finitely generated integral domain over k that is the quotient of a polynomial ring arises as the coordinate ring of a unique irreducible affine variety, up to isomorphism.^[18]^[52] Hilbert's Nullstellensatz establishes a precise correspondence between ideals in the polynomial ring and geometric objects. Specifically, over an algebraically closed field, the maximal ideals of k[V] correspond bijectively to the points of V, as each maximal ideal is of the form \mathfrak{m}_p = \{f \in k[V] \mid f(p) = 0\} for p \in V. Moreover, for any ideal J \subseteq k[x_1, \dots, x_n], the radical \sqrt{J} equals the ideal of all polynomials vanishing on the variety V(J), ensuring that radical ideals define varieties and vice versa. This theorem underpins the ideal-variety duality, allowing algebraic manipulations of ideals to reflect geometric properties of varieties.^[53]^[19] The Krull dimension of the coordinate ring k[V], defined as the supremum of lengths of chains of prime ideals, equals the dimension of the variety V, which is the Krull dimension of its defining ring. This algebraic dimension captures the geometric notion of transcendence degree of the function field over k, providing a measure of the "size" of V. For instance, the affine line has dimension 1, with k having Krull dimension 1.^[54]^[55] An integral domain is integrally closed in its field of fractions if and only if the corresponding affine variety is normal, meaning it is nonsingular in codimension 1 or, more precisely, that its local rings at every point are integrally closed domains. Normalization resolves singularities by adjoining integral elements, yielding a normal variety birational to the original. For example, the ring k[x, y]/(xy) is not an integral domain, as xy = 0 implies the images of x and y are zero divisors; geometrically, this corresponds to the reducible curve consisting of the union of the x-axis and y-axis in \mathbb{A}^2_k.^[56]^[57]^[58]

Number Theory Connections

In algebraic number theory, the ring of integers \mathcal{O}_K of a number field K plays a central role as an integral domain that is always a Dedekind domain. This structure ensures that \mathcal{O}_K is Noetherian, integrally closed in its field of fractions K, and of Krull dimension 1.^[59]^[60] A Dedekind domain is characterized by the property that every nonzero prime ideal is maximal, reflecting its one-dimensional nature. Moreover, every nonzero proper ideal in a Dedekind domain admits a unique factorization into a product of prime ideals, up to ordering and units. This ideal-theoretic unique factorization compensates for the potential failure of unique factorization in elements, which is a hallmark of principal ideal domains but not all integral domains.^[60] The ideal class group of a Dedekind domain quantifies the extent to which unique element factorization fails, defined as the quotient of the group of fractional ideals by the subgroup of principal fractional ideals. In the context of \mathcal{O}_K, this group is finite, and its order, known as the class number, measures the deviation from being a principal ideal domain. For instance, the ring of integers \mathbb{Z} of the rational field \mathbb{Q} is a Dedekind domain with trivial ideal class group, as it is a principal ideal domain.^[61]^[59] Examples abound in quadratic number fields K = \mathbb{Q}(\sqrt{d}) for square-free integers d. The ring of integers is \mathbb{Z}[\sqrt{d}] if d \equiv 2, 3 \pmod{4}, and \mathbb{Z}\left[\frac{1 + \sqrt{d}}{2}\right] if d \equiv 1 \pmod{4}; both are Dedekind domains. In imaginary quadratic fields like \mathbb{Q}(\sqrt{-5}), the ideal class group has order 2, illustrating non-principal ideals such as (2, 1 + \sqrt{-5}). In contrast, real quadratic fields like \mathbb{Q}(\sqrt{5}) often have class number 1.^[59] The discriminant of \mathcal{O}_K further connects integral domains to the arithmetic of extensions, defined as the determinant of the trace form on a \mathbb{Z}-basis of \mathcal{O}_K. It encodes information about ramification: a prime p ramifies in K/\mathbb{Q} if and only if p divides the discriminant, meaning the prime ideal factorization of (p) in \mathcal{O}_K involves repeated factors. For quadratic fields, the discriminant is $4d or d depending on d \pmod{4}, directly indicating ramified primes.^[62]^[59]

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[PDF] Ring Fundamentals - UT Math
It is possible for the product of two nonzero matrices to be zero, so that Mn(R) is not an integral domain. (To generate a large class of examples, let Eij ...
[21]
[PDF] Name: Solutions Final Exam Instructions. Answer each ... - Math@LSU
Dec 2, 2018 · (Recall that a nonzero element a in a ring R is a zero-divisor if there is a nonzero b ∈ R with ab = 0.) ▷ Solution. Z4 = Z/4Z is a factor ring ...
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[PDF] Chapter 3, Rings Definitions and examples. We now have several ...
Example. Let R be the ring Z × Z = { (n, m) | n, m ∈ Z}. Note that (1, 0)(0, 1) = (0, 0) = 0R, so R is not an integral domain. For the same reason, no product ...
[23]
[PDF] Math 403 Chapter 13: Integral Domains and Fields 1. Introduction
(b) Definition: A commutative ring with a unity is an integral domain if it has no zero- divisors. In other words a commutative ring with unity is an integral ...
[24]
[PDF] 31 Prime elements
31.3 Definition. Let R is an integral domain. An element a ∈ R is a prime element if p 6= 0, p is a non-unit and if a | bc then either a | b or a | c. 31.4 ...
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[PDF] Lecture #21 of 38 ∼ March 10, 2021 - Math 4527 (Number Theory 2)
Definition Let R be an integral domain. A nonzero element p ∈ R is prime if p is nonzero and not a unit, and for any a,b ∈ R, if p|ab then p|a or p|b. ...
[26]
[PDF] Modern Algebra
Mar 26, 2024 · Let p and c be nonzero elements in an integral domain. R. (i) p is prime if and only if (p) is a nonzero prime ideal. Proof. (i) Let p be prime.
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[PDF] Math 403 Chapter 18: Irreducibles, Associates, Primes, UFDs
(d) Theorem: In an integral domain every prime is irreducible. Proof: Suppose a ∈ D is prime and a = bc. We claim that one of b, c is a unit. Since a = bc ...
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[PDF] Divisibility and Factorial Rings Definition. Irreducible element in an ...
(2) The principal ideal generated by an irreducible element is prime. (3) All non-trivial prime ideals are maximal. An integral domain satisfying (1) and (2) ...
[29]
[PDF] Primes and absolutely or non-absolutely irreducible elements ... - arXiv
Let H be a cancellative commutative monoid. (i) An irreducible element r ∈ H is called absolutely irreducible. (or, a strong atom), if for all natural numbers ...
[30]
[PDF] GAUSSIAN INTEGERS Contents 1. Principal Ideal Domain and ...
A non-zero non-unit element p in an integral domain is irreducible if it is not the product of two non-zero units. In the ring of integers Z, prime and ...
[31]
10.27 Examples of spectra of rings - Stacks Project
Now, I claim that if f, g are irreducible polynomials in k[x, y] that are not associates, then (f, g) \cap k[x] \neq 0. To do this, it is enough to show that f ...
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[PDF] Integral Domains - Assets - Cambridge University Press
1.1 Integral Domains. In this chapter we recall the definition and properties of an integral domain and develop the concept of divisibility in such a domain.
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[PDF] Algebra: ring and field theory - UCSD Math
domains. Lemma 3.3.6 (Cancellation law). Suppose D is an integral domain. Then for every non-zero a ∈ D and b, c ∈ D, ab = ac implies b = c. Proof. Since ...
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[PDF] More ring theory | MIT ESP
We first show that a quotient by a prime ideal is an integral domain. We have that everything in the quotient ring A/I that is equal to zero is an element ...
[35]
[PDF] Math101b_notesB1: Integrality - Brandeis
A domain (= entire ring) is called integrally closed if it is integrally closed in its fraction field. The last theorem shows that Z is integrally closed.
[36]
[PDF] worksheet on artinian rings with proofs
Let R be an Artinian integral domain. Then R is a field. Proof. Let x ∈ R be nonzero and consider (x) ⊇ (x2) ⊇ ... which must stabilize because R is. Artinian.
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[PDF] 14. Field of fractions If R is an integral domain we have ... - UCSD Math
For every integral domain R we would like to construct a field F which contains R (or at least a copy of R). The rational numbers Q are constructed from the ...
[38]
[PDF] Homework #5 Solutions
But R is a subring of R[x] so, as above, these elements suffice to show that the characteristic of R[x] cannot be less than n. Hence,. charR[x] = n = charR ...
[39]
[PDF] Math 103B - HW-2 (solution)
Feb 7, 2020 · Thus. A is a subdomain of D. 27 Show that the characteristic of a subdomain of an integral domain D is equal to the characteristic of D. Proof.<|separator|>
[40]
Integral Domains - Department of Mathematics at UTSA
Dec 19, 2021 · In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero.Definition · Non-examples · Divisibility, prime elements... · Properties
[41]
[PDF] BEZOUT DOMAINS AND ELLIPTIC CURVES
A Bezout domain is an integral domain such that any finitely generated ideal is principal. Clearly a noetherian Bezout domain is a PID. Examples of non ...
[42]
[PDF] Bézout rings with almost stable range 1
The notion of an adequate domain was originally defined by Helmer [10]. The ring of entire functions on the complex plane is an adequate Bézout domain.
[43]
[PDF] Bézout Domains and Elementary Divisor Domains
Apr 18, 2023 · Therefore, another way to describe a Bézout domain is it is a domain that is both Schreier and Prüfer. Remark 2.1.4. There are many equivalent ...
[44]
Does a UFD imply Euclidean Domain? - Mathematics Stack Exchange
Mar 2, 2020 · I'm struggling to get my head around the relationship between UFD, PID and Euclidean Domain. I've seen in a theorem in my notes that Euclidean Domain ⇒ PID ⇒ ...
[45]
[PDF] Rings of fractions the hard way José Felipe Voloch Let R ... - UT Math
Now we prove the usual fact that integral domains have a field of fractions. Theorem 2. If R is an integral domain and D ⊂ R \ {0} then the ring of fractions of ...
[46]
[PDF] 36 Rings of fractions
1) If R is an integral domain and S = R − {0} then the ring S−1R is a field. In this case S−1R is called the field of fractions of R. 2) If I is a prime ...
[47]
[PDF] January 26, 2022 1. Rings, ideals, fields A r
Now, although an integral domain R can in general fail to be a field, an isomorphic copy of R will always be contained in a field called its field of fractions.<|control11|><|separator|>
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[PDF] Math103b, lecture 5 - UCSD Math
Theorem 4 (Universal Property of Field of Fractions) Suppose. D is an integral domain and Q(D) is its field of fractions. Suppose. F is a field and θ : D → F ...
[49]
[PDF] 18.703 Modern Algebra, Field of Fractions - MIT OpenCourseWare
phism and that it satisfies the given universal property. D. Example 17.3. If we take R = Z, then of course the field of fractions is isomorphic to Q. If R ...
[50]
Visual Algebra - Mathematical and Statistical Sciences | Clemson ...
We can mimic this construction in any integral domain. Definition. Given an integral domain R, its field of fractions is the set. R × R∗ = (a, b) | a, b ∈ R ...
[51]
[PDF] worksheet #2 – math 6310 fall 2017
If R is an integral domain, W = R \ {0}, and K = W-1R is the field of fractions, show that K ... This is called the universal property of localization. 4.
[52]
[PDF] Affine Varieties
An affine algebraic set X is irreducible if and only if O(X) is an integral domain. Proof. Embed X into affine space An, so O(X) = F[X1,ททท,Xn]/a for some ...
[53]
[PDF] Ideal/variety correspondence and Hilbert's nullstellensatz
(1) Every maximal ideal of R is of the form mP = (X1 − r1,..., Xn − rn) = I(P) for some point P = (r1,..., rn) ∈ kn. (2) If J ⊊ R is a proper ideal then V(J) ≠ ...<|control11|><|separator|>
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[PDF] Dimension of a Variety - DIMACS REU
Jul 14, 2020 · We define I(A) = {f ∈ K[x1,x2, ...,xn]|f (a)=0∀a ∈ A}. 2. The coordinate ring of a variety is then K[x1, ...,xn]/I(A). 3. Given a ring R, ...
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Coordinate Ring -- from Wolfram MathWorld
f(x_1,x_2)=0 in the affine plane K^2 is K[x_1,x_2]/<f(x_1,x_2)> . In general, the Krull dimension of ring K[V] is equal to the dimension of V as a closed ...
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[PDF] /usr/local/bin/dvialw Normal - University of Utah Math Dept.
Definition: An affine variety X is normal if C[X] is integrally closed. Examples: (a) Cn is normal, since C[x1, ..., xn] is a UFD. (b) C[X] is not normal if ...
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[PDF] 1 Irreducible and integral schemes - Kiran S. Kedlaya
A scheme is irreducible if its underlying topological space is irreducible. A scheme is integral if every open affine subscheme is the spectrum of an integral ...
[58]
[PDF] Algebraic Geometry Part III Catch-up Workshop 2015
Jun 6, 2016 · We saw that Z(XY ) ⊂ A2 is reducible. The coordinate ring is k[X, Y ]/(XY ), which is not an integral domain, since it contains the non-zero ...
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[PDF] Algebraic Number Theory - James Milne
Feb 11, 2008 · ˘ ideals in Dedekind domains factor uniquely into products of prime ideals, and. ˘ rings of integers in number fields are Dedekind domains,.
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[PDF] Properties of Dedekind Domains and Factorization of Ideals
Sep 15, 2021 · The fact that we have unique factorization of ideals implies that it is enough to show that every prime ideal is principal. Let p be a nonzero ...
[61]
[PDF] Ideal classes and Kronecker bound - Keith Conrad
Ideal class groups of number fields are fundamental objects in number theory. We will prove the ideal class group of every number. 1. Page 2. 2.
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[PDF] Discriminants and ramified primes - Keith Conrad
The discriminant of OK is discZ(OK). The next lemma says reduction modulo p commutes (in a suitable sense) with the formation of discriminants. Lemma 3.3.