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Primary ideal

In , a primary ideal is a proper ideal Q of a A such that whenever the product xy \in Q for elements x, y \in A, either x \in Q or y^n \in Q for some positive integer n. This condition is equivalent to the A/Q having the property that every zero-divisor is . Primary ideals generalize , as every prime ideal satisfies this property (taking n = 1), but the converse does not hold; for instance, in the \mathbb{Z}, the ideal (p^n) generated by the n-th power of a prime p is primary with radical (p), which is . Key properties of primary ideals include the fact that the \sqrt{Q} = \{ r \in A \mid r^k \in Q \text{ for some } k \geq 1 \} of a primary ideal Q is a , and Q is called \mathfrak{p}-primary if \sqrt{Q} = \mathfrak{p}. This association links primary ideals to the prime ideals containing them, facilitating the study of ideal structure in rings. In polynomial rings over fields, such as k[x, y], examples like (x, y^2) illustrate primary ideals whose radicals are maximal ideals like (x, y). Primary ideals are neither necessarily prime nor maximal, but they form the building blocks for decomposing more complex ideals. Primary ideals were introduced by in 1905 in the context of ideals in rings, where he used them to establish results. later generalized these ideas in 1921, proving the Lasker-Noether theorem, which states that in Noetherian rings, every proper ideal admits a finite as an intersection of primary ideals, with the associated primes (radicals of the primary components) being unique in minimal decompositions. This theorem is foundational in , enabling the analysis of modules, varieties, and schemes, and underscoring the role of primary ideals in bridging arithmetic and geometric interpretations of .

Definition and motivation

Definition

In commutative algebra, a commutative ring R with identity is an abelian group under addition equipped with a multiplication operation that is associative, commutative, and distributive over addition, and possesses a multiplicative $1_R \neq 0. An I of R is a subset that is an additive of R and absorbs multiplication by elements of R, meaning that if a \in I and r \in R, then ra \in I. The zero ideal of R, denoted (0), consists solely of the element $0. An ideal Q of R is primary if Q \neq R and whenever ab \in Q for a, b \in R, then either a \in Q or b^n \in Q for some positive integer n. Equivalently, Q is primary if and only if the R/Q is nonzero and every zero-divisor in R/Q is . Prime ideals form a special case of primary ideals, where the condition holds with n = 1. The of Q, denoted \mathfrak{p} = \sqrt{Q}, is the set of all elements x \in R such that x^n \in Q for some positive n.

Historical development

The concept of primary ideals emerged in the early as part of efforts to understand the structure of ideals in rings, particularly in relation to solving systems of equations. introduced primary ideals in 1905 while studying modules and ideals in rings over fields, motivated by geometric considerations in following of 1893. In his seminal work, Lasker established that every in such rings admits a into primary ideals, providing an early version of what would later become the Lasker-Noether theorem. Building on David Hilbert's basis theorem from 1890, which proved that polynomial rings over Noetherian rings are Noetherian, extended these ideas to a broader algebraic framework in the 1920s. In her 1921 paper "Idealtheorie in Ringbereichen," Noether developed the theory of ideals in commutative rings with ascending chain conditions (now known as Noetherian rings), defining primary ideals in this abstract setting and proving their role in the unique factorization of ideals up to radicals. This work unified the study of polynomial rings and rings of integers, shifting focus from specific cases to general structural properties. Wolfgang Krull further refined the theory in the 1930s, incorporating primary ideals into a comprehensive treatment of ideal theory in commutative rings. In his 1935 monograph "Idealtheorie," Krull explored intersections and dimensions of ideals, enhancing the understanding of primary components and their associated primes, while addressing more general rings without finite generation assumptions in earlier papers. These contributions solidified the abstract foundations laid by Noether. Following , the concept evolved within the burgeoning field of abstract , influenced by the Bourbaki group's emphasis on structural rigor, moving away from initial geometric motivations toward pure ring-theoretic developments. This progression foreshadowed the central role of primary ideals in the theorem.

Properties and characterizations

Fundamental properties

A primary ideal Q in a commutative ring R has the property that its \sqrt{Q} = \{ r \in R \mid r^n \in Q \ \text{for some positive integer} \ n \} is a , denoted \mathfrak{p}, and Q is said to be \mathfrak{p}-primary. Consequently, if Q is primary, the R/Q is a primary ring, meaning that its nilradical coincides with a prime ideal. This is equivalent to the condition that every zero-divisor in R/[Q](/page/Q) is . The of primary ideals need not be primary in general; however, the finite of \mathfrak{p}-primary ideals is again \mathfrak{p}-primary. In Noetherian rings, if [Q](/page/Q) is primary with \mathfrak{p}, then \mathfrak{p}^k \subseteq [Q](/page/Q) for some positive k.

Equivalent conditions

In , an Q in a R with identity is primary if and only if whenever ab \in Q for a, b \in R, either a \in Q or b \in [\sqrt{Q}](/page/Radical), where \sqrt{Q} denotes the of Q. This condition is equivalent to the classical power condition that ab \in Q implies a \in Q or b^n \in Q for some positive n. To see the equivalence, first note that the power condition implies the radical condition, since b^n \in Q entails b \in \sqrt{Q}. Conversely, suppose the radical condition holds and ab \in Q with a \notin Q. In the quotient ring R/Q, the images \overline{a} and \overline{b} satisfy \overline{a} \overline{b} = 0 with \overline{a} \neq 0, so \overline{a} is a zero-divisor; by the radical condition, \overline{b} \in \sqrt{0}, the nilradical of R/Q, hence b^n \in Q for some n. Thus, every zero-divisor in R/Q is nilpotent. A related reformulation views Q through the quotient module R/Q. The ideal Q is primary if and only if the R-module R/Q is primary, meaning that every zero-divisor on R/Q (i.e., any r \in R such that multiplication by r sends some nonzero element of R/Q to zero) acts nilpotently on R/Q. In other words, for any r \in R with r (R/Q) \neq R/Q but r x = 0 for some x \neq 0 in R/Q, there exists n > 0 such that r^n (R/Q) = 0, or equivalently r^n \in Q. This module-theoretic perspective generalizes the ideal case and aligns with the zero-divisor condition above. Further, Q is primary if and only if the R-module R/Q has exactly one associated prime ideal, which is the nilradical \sqrt{0} of R/Q (equivalently, \sqrt{Q}/Q). The associated primes of R/Q are the prime ideals \mathfrak{p} of R containing Q such that \mathfrak{p} = \mathrm{Ann}_R(\overline{x}) for some nonzero \overline{x} \in R/Q. Under the zero-divisor nilpotency, the only such prime is \sqrt{Q}, which must itself be prime (as referenced in fundamental properties). This characterization is particularly useful in , where the associated primes distinguish the components. Finally, Q is primary if and only if its localization at the multiplicative set S = R \setminus \sqrt{Q} is a primary ideal in the localized ring S^{-1}R. Since \sqrt{Q} is prime, S avoids \sqrt{Q}, and the localized ideal S^{-1}Q satisfies the zero-divisor nilpotency in S^{-1}R / S^{-1}Q, preserving the primary nature. This localization property facilitates proofs involving local rings and supports generalizations to non-Noetherian settings.

Examples

Basic examples

In the ring of integers \mathbb{Z}, the principal ideals (p^n) generated by the nth power of a p, where n \geq 1, are primary ideals with associated prime ideal (p). For instance, the ideal (8) = (2^3) is $2-primary, as its is (2) and it satisfies the primary condition: if ab \in (8) and a \notin (8), then some power of b lies in (8). In the k over a k, the principal ideals (x^n) for n \geq 1 are primary with (x). These ideals illustrate how powers of the zero ideal's generator yield primary ideals in a . Additionally, maximal ideals such as (x - a) for a \in k are prime and hence primary. A basic example of a primary ideal that is not prime arises in \mathbb{Z}, where the ideal (2, x^2) is primary with radical (2, x), a maximal ideal. To see it is not prime, note that x \cdot x = x^2 \in (2, x^2), but x \notin (2, x^2) since no combination of generators produces a linear term with odd coefficient. In contrast, while the zero ideal (0) in \mathbb{Z} is primary (in fact, prime), the zero ideal in the quotient ring k[x, y] / (xy) is not primary. Here, x y = 0 \in (0), but neither x nor y is in (0), and no positive power of x or y is zero. This highlights how the primary property can fail in rings with zero divisors.

Examples in specific rings

In polynomial rings over a field, such as k[x_1, \dots, x_n], examples of primary ideals include certain monomial ideals where, in the quotient ring, the image of each variable is either regular or nilpotent. For instance, in k[x, y], the ideal (x, y^2) is primary with radical the maximal ideal (x, y), as the quotient k[x, y] / (x, y^2) \cong k / (y^2) has all zero-divisors nilpotent. Similarly, (x^2, y) in k[x, y] is primary to (x, y), appearing as a component in the primary decomposition of ideals like (x^2, x y). In the polynomial ring \mathbb{Z}, primary ideals can be constructed as (p^n, f(x)), where p is a prime number and f(x) is a polynomial irreducible modulo p. For example, the ideal (9, 2x + 1) is primary, since $2x + 1 is irreducible over \mathbb{F}_3 and the radical is the prime ideal (3, 2x + 1). In power series rings like k[], which is a principal ideal domain and discrete valuation ring, the nonzero ideals are of the form (x^n) for n \geq 1, each of which is primary to the maximal ideal (x). Geometrically, in over an , a primary ideal corresponds to an irreducible equipped with a non-reduced structure that is uniform across the entire , without additional embedded components on proper subvarieties.

Primary decomposition

The primary decomposition theorem

In a R, the theorem guarantees that every ideal I admits a , meaning I can be written as a finite I = Q_1 \cap \cdots \cap Q_k, where each Q_i is a primary ideal. This result, originally established by in her foundational work on ideal theory, extends earlier ideas from and forms a cornerstone of by providing a way to "factor" ideals into primary components analogous to prime in integers. The theorem relies on the ascending chain condition inherent to s, ensuring that such decompositions are finite. The existence of this decomposition follows from two key lemmas. First, every ideal in a Noetherian ring is a finite of irreducible ideals, obtained by iteratively decomposing non-irreducible ideals using the Noetherian property to avoid infinite descending chains of ideals. Second, in Noetherian rings, every irreducible ideal is , as any product ab \in Q with a \notin Q implies some power of b lies in Q, leveraging the maximality of the . Combining these, any ideal yields a primary decomposition. A is called minimal or irredundant if the ideals p_i = \sqrt{Q_i} are distinct and no component is superfluous, meaning for each i, Q_i \not\subseteq \bigcup_{j \neq i} Q_j. To obtain a minimal from a general one, redundant primary ideals—those contained in the intersection of the others—are simply removed, resulting in a representation I = \bigcap_{i=1}^k Q_i where the p_i may include both minimal (isolated) and primes over I. primes arise when a primary component is "deeper" in the , not minimal among the s. This refinement preserves the intersection while highlighting the essential prime structure associated to I.

Associated primes and irredundancy

In the context of , the associated primes of an ideal I in a R are defined as the set \operatorname{Ass}(I) = \{ \mathfrak{p} \mid \mathfrak{p} = \sqrt{Q} \text{ for some primary ideal } Q \text{ appearing in a primary decomposition of } I \}. This set captures the prime ideals that "underlie" the primary components and plays a crucial role in understanding the structure of I. The associated primes are precisely the annihilators of nonzero elements in the R/I, providing an intrinsic characterization independent of any particular decomposition. A key theorem states that the set of associated primes \operatorname{Ass}(I) is independent of the choice of primary decomposition of I. Moreover, among the associated primes, the minimal elements with respect to inclusion are exactly the minimal prime ideals over I, while the non-minimal ones are known as primes. This uniqueness ensures that while primary decompositions themselves may not be unique, the "skeleton" formed by their radicals is . A primary decomposition I = \bigcap_{i=1}^n Q_i, where each Q_i is primary, is called irredundant if for every i, Q_i \not\supseteq \bigcap_{j \neq i} Q_j. Irredundancy eliminates superfluous components and guarantees that the associated primes are exactly the distinct radicals \sqrt{Q_i} without repetition. In an irredundant decomposition, no primary component can be omitted without altering the , reflecting the essential structure of I. For example, consider the I = (x^2, xy) in the k[x,y] over a k. One irredundant is I = (x) \cap (x^2, y), where (x) is prime (hence primary) with (x), and (x^2, y) is (x,y)-primary with (x,y). The associated primes are thus \operatorname{Ass}(I) = \{(x), (x,y)\}, with (x) minimal and (x,y) embedded. This is irredundant because neither component contains the other.

Relations to other concepts

Comparison with prime ideals

A prime ideal \mathfrak{p} in a commutative ring R is defined as a proper ideal such that for any a, b \in R, if ab \in \mathfrak{p}, then a \in \mathfrak{p} or b \in \mathfrak{p}. This corresponds to the case n=1 in the definition of a primary ideal, where an ideal \mathfrak{q} satisfies: if ab \in \mathfrak{q}, then either a \in \mathfrak{q} or b^n \in \mathfrak{q} for some positive n. Thus, every is primary, as the prime implies the primary by taking n=1. However, the converse does not hold; there exist primary ideals that are not prime. For instance, in the k over a k, the ideal (x^2) is primary but not prime. To see it is not prime, note that x \cdot x \in (x^2) yet x \notin (x^2). It is primary because if f(x) g(x) \in (x^2), the multiplicity of the root at x=0 in the product is at least 2, so at least one of f or g has multiplicity at least 2 (hence in (x^2)) or the other has multiplicity at least 1 raised to some power still ensuring the condition. A key distinction is that the radical of a primary ideal \mathfrak{q}, denoted \sqrt{\mathfrak{q}}, is prime, and for prime ideals \mathfrak{p}, \sqrt{\mathfrak{p}} = \mathfrak{p}, reflecting that primes are radical while primaries may not be. Primary ideals thus generalize primes by allowing "higher powers" to account for multiplicity in zero-divisors. Geometrically, in the of algebraic , prime ideals correspond to irreducible subvarieties in the \operatorname{Spec}(R). Primary ideals, by contrast, capture more structure: a \mathfrak{p}-primary ideal corresponds to a primary component associated to the irreducible variety V(\mathfrak{p}), incorporating embedded components or multiplicity along that variety. This allows primary ideals to describe non-irreducible phenomena, such as points with higher multiplicity or embedded points in the support of modules, where the has zero-divisors but no zero-divisors outside the nilradical.

Role in Noetherian rings

In Noetherian rings, the primary decomposition theorem guarantees that every admits a finite , a cornerstone result known as the Lasker–Noether theorem. This finite decomposition is essential for analyzing the structure of ideals, as it expresses any ideal I as I = Q_1 \cap \cdots \cap Q_n where each Q_i is primary, and the associated primes \sqrt{Q_i} are distinct in minimal decompositions. The existence of such decompositions enables key applications, including proofs of ; for instance, utilized to establish an early version of the theorem, linking algebraic ideals to geometric varieties over algebraically closed fields. Computationally, primary ideals play a vital role in over fields, where algorithms leverage Gröbner bases to compute decompositions explicitly. A seminal method is the Gianni–Trager–Zacharias algorithm, which decomposes any ideal in a over a into primary components by first finding the and then isolating associated primes. This approach has been implemented in systems like SINGULAR, facilitating practical computations in and elimination theory. Primary decompositions also determine the support of finitely generated modules over Noetherian rings, defined as the set of primes P such that the localization M_P \neq 0. For a module M with submodule N = \bigcap Q_i in primary decomposition, the support of M/N is the union of the supports of the primary components, with associated primes \operatorname{Ass}(M/N) = \{\sqrt{Q_i}\}. The length of the minimal decomposition relates to the module's structure, and the dimensions of the associated primes govern the of the support, where the codimension of an associated prime P is \dim R - \dim R/P, providing insights into the geometric dimension of varieties associated to the ideal. While primary ideals are well-behaved in Noetherian settings, generalizations to non-Noetherian or non-commutative rings face significant limitations; for example, submodules may lack primary decompositions, and associated primes may not exist in the classical sense, complicating structural analysis.