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Commutative algebra

Commutative algebra is the branch of that studies commutative rings with unity, their ideals, modules, and related structures, focusing on properties such as localization, tensor products, and the . It originated from efforts in and , where rings like the integers \mathbb{Z} and polynomial rings k[x_1, \dots, x_n] over a k serve as fundamental examples. The field emphasizes Noetherian rings, unique factorization domains, and Dedekind domains, providing tools to analyze factorization and dimension in algebraic varieties. Central concepts include prime ideals, which generalize both arithmetic primes and geometric points, and localization at primes, which allows "zooming in" on properties analogous to differential geometry's local analysis. Commutative algebra forms the algebraic foundation for modern algebraic geometry, particularly through Grothendieck's theory of schemes, where the spectrum \operatorname{Spec}(R) of a ring R endows it with a geometric structure. It also connects to homological algebra via Ext and Tor functors, flat modules, and notions like Cohen-Macaulay rings and regularity, enabling the study of smoothness and étaleness in morphisms. These tools have profound applications in number theory, such as class field theory, and in solving systems of polynomial equations.

Introduction

Definition and Scope

Commutative algebra is the branch of dedicated to the study of and the modules over them. A is defined as a R with a multiplicative identity (unity) element 1 such that is commutative, meaning ab = ba for all a, b \in R. This structure builds on the general notion of a , which requires addition to form an and to be associative and distributive over addition, but commutative algebra restricts attention to those rings where the multiplicative operation satisfies the commutativity axiom. The scope of commutative algebra centers on the algebraic structures arising from , particularly ideals, modules, and their homological properties, such as projective and injective dimensions. Ideals in a R are subsets closed under addition and absorption by ring elements, enabling the formation of quotient rings R/I. Modules over generalize vector spaces, allowing linear algebra techniques to be applied in settings without a , and homological tools like Ext and functors reveal deeper relationships between these objects. This focus distinguishes commutative algebra from broader , which encompasses non-commutative rings where multiplication does not commute, leading to different behaviors in ideals and representations. Basic examples of commutative rings include the integers \mathbb{Z}, which form a under standard addition and ; polynomial rings k[x_1, \dots, x_n] over a k, which model multivariate functions and possess desirable finiteness properties; and quotient rings like \mathbb{Z}/n\mathbb{Z}, which capture . In contrast, rings of n \times n matrices over a field with n \geq 2 are non-commutative, as matrix multiplication generally fails to commute, and thus fall outside the purview of commutative algebra. Commutative algebra underpins algebraic geometry, where rings correspond to affine schemes that geometrize algebraic varieties.

Motivations and Importance

Commutative algebra emerged historically to address key challenges in and , particularly through David Hilbert's development of the Nullstellensatz in , which established a profound correspondence between in polynomial rings over algebraically closed fields and geometric varieties defined by those polynomials. This theorem provided algebraic tools to determine whether systems of polynomial equations have solutions, bridging abstract with the study of solution sets in , thereby motivating the systematic exploration of commutative rings to model geometric objects. For instance, the of a by an I captures the zero set V(I) of polynomials generating I, offering a concrete way to represent varieties as algebraic structures. In , commutative algebra gained motivation from efforts to solve Diophantine equations, where introduced ideals in the 1870s to extend unique beyond , enabling the factorization of elements in rings of algebraic integers and thus tackling equations like those arising in quadratic fields. This approach resolved issues in determining solvability, such as showing that certain equations like y² = x³ - 5 lack rational solutions in specific extensions due to the structure of ideals. The resulting , which quantifies the failure of principal ideals to generate all invertible ideals in Dedekind domains, became essential for investigations, including criteria for the existence of solutions in algebraic number fields. The importance of commutative algebra lies in its role as a foundational bridge between and concrete applications in and arithmetic, providing the machinery to analyze systems and solutions rigorously. In modern contexts, it underpins scheme theory in , where schemes—constructed from commutative rings via the —unify classical varieties with more general objects, facilitating the study of families of geometric structures over arbitrary bases and resolving longstanding problems like the . Similarly, in , concepts like the inform the arithmetic of fields, influencing results on Diophantine solvability and the distribution of primes.

Basic Concepts

Commutative Rings and Ideals

A is an consisting of a set R equipped with two binary operations, and , satisfying the axioms of an under , associativity and distributivity of over , and commutativity of : for all a, b \in R, ab = ba. Typically, commutative rings are assumed to have a , denoted $1, such that $1 \cdot r = r \cdot 1 = r for all r \in R; this is called a with . Subrings of a R with are subsets S \subseteq R that are themselves with under the induced operations and containing the same $1. A between R and S with is a \phi: R \to S that preserves , , and the : \phi(a + b) = \phi(a) + \phi(b), \phi(ab) = \phi(a)\phi(b), and \phi(1_R) = 1_S. An in a R is a nonempty additive I \subseteq R such that for all r \in R and i \in I, ri \in I ( under multiplication by elements). The of two ideals I and J in R is the set I + J = \{i + j \mid i \in I, j \in J\}, which is itself an , as it inherits closure under addition and from I and J. A is an generated by a single element a \in R, denoted (a) = \{ra \mid r \in R\}. A prime ideal P is a proper ideal such that if ab \in P for a, b \in R, then a \in P or b \in P; equivalently, the quotient R/P is an integral domain (a with unity and no zero divisors). A maximal ideal M is a proper ideal not contained in any larger proper ideal; in with unity, every maximal ideal is prime, and R/M is a field. Given an I in a R, the R/I is the set of cosets \{r + I \mid r \in R\} with operations (r_1 + I) + (r_2 + I) = (r_1 + r_2) + I and (r_1 + I)(r_2 + I) = (r_1 r_2) + I, forming a commutative ring with unity $1 + I. The correspondence theorem states that there is a between the ideals of R containing I and the ideals of R/I, given by J \mapsto J/I for J \supseteq I, preserving inclusion and sums. Examples illustrate these concepts prominently. The ring of integers \mathbb{Z} under usual addition and multiplication is a commutative ring with unity, and every ideal in \mathbb{Z} is principal, generated by some nonnegative integer n, making \mathbb{Z} a principal ideal domain (PID). For a field k, the polynomial ring k is a Euclidean domain with the degree function as the Euclidean norm, allowing division algorithm, and thus every ideal is principal. The radical of an ideal I in a commutative ring R, denoted \sqrt{I}, is the set \{r \in R \mid r^n \in I \text{ for some positive integer } n\}, which is itself an ideal containing I.

Modules over Commutative Rings

In commutative algebra, provide a framework for studying linear structures over s, generalizing vector spaces from linear algebra. Let [R](/page/R) be a with identity. An [R](/page/R)- [M](/page/M) is an (written additively) equipped with a map [R](/page/R) \times [M](/page/M) \to [M](/page/M), denoted (r, m) \mapsto r m, satisfying the axioms: r(m + n) = r m + r n, (r + s) m = r m + s m, (r s) m = r (s m), and [1](/page/1) \cdot m = m for all r, s \in [R](/page/R) and m, n \in [M](/page/M). The commutativity of [R](/page/R) ensures that the scalar action commutes, meaning r (s m) = s (r m) for all r, s \in [R](/page/R) and m \in [M](/page/M), which simplifies the endomorphism ring structure and allows to behave analogously to vector spaces over fields while accommodating zero divisors and non-invertible elements. This symmetry facilitates key constructions like tensor products and localizations, central to commutative algebra. Submodules and quotient modules extend the subgroup and quotient group concepts to the module setting. A submodule N of an R-module M is a of M that is closed under by elements of R, i.e., r n \in N for all r \in R and n \in N. If N is a submodule, the M/N is the set of cosets m + N with induced addition and : (m + N) + (m' + N) = (m + m') + N and r (m + N) = r m + N. Exact sequences capture relationships among modules via homomorphisms: a sequence \cdots \to M_{i-1} \xrightarrow{f_{i-1}} M_i \xrightarrow{f_i} M_{i+1} \to \cdots of R-modules and R-linear maps is exact at M_i if the image of f_{i-1} equals the kernel of f_i, i.e., \operatorname{im} f_{i-1} = \ker f_i. Commutativity ensures these structures inherit the ring's properties without additional complications from non-commuting scalars. Free and projective modules represent "simplest" types of modules in this context. A R-module is one that possesses a basis, meaning it is isomorphic to a of copies of R, such as R^{(I)} = \bigoplus_{i \in I} R for some I, where elements are finite linear combinations \sum r_i e_i with e_i the vectors. A P is a direct summand of a , i.e., there exists a F and modules Q, S such that F \cong P \oplus Q and P \cong S as R-modules; equivalently, the functor \operatorname{Hom}_R(P, -) is exact. Over commutative rings, often coincide with in important cases, like principal ideal domains, but differ in general, as seen in non-free projectives over rings. The tensor product operation combines two R-modules M and N into a new module M \otimes_R N, which is the abelian group generated by symbols m \otimes n (for m \in M, n \in N) subject to bilinearity relations: (m + m') \otimes n = m \otimes n + m' \otimes n, m \otimes (n + n') = m \otimes n + m \otimes n', and (r m) \otimes n = m \otimes (r n) = r (m \otimes n) for r \in R. Commutativity of R makes this scalar action well-defined and symmetric, enabling M \otimes_R N to capture bilinear maps from M \times N. The annihilator ideal of a module M, denoted \operatorname{Ann}(M) = \{ r \in R \mid r m = 0 \ \forall m \in M \}, measures the "kernel" of the action of R on M and is indeed an ideal due to commutativity. Representative examples illustrate these concepts. The ring R itself is a free R-module of rank one, with basis \{1\}. Any ideal I \subseteq R forms a submodule of R, and the quotient R/I is an R-module via the induced . If R is a , then R-modules are precisely vector spaces over R. For instance, over R = \mathbb{Z}, the integers as a ring, modules are abelian groups, ideals like $2\mathbb{Z} are submodules, and tensor products like \mathbb{Z}/2\mathbb{Z} \otimes_\mathbb{Z} \mathbb{Z}/3\mathbb{Z} \cong \mathbb{Z}/\gcd(2,3)\mathbb{Z} = 0 highlight torsion interactions.

Historical Development

Early Foundations

The foundations of commutative algebra trace back to mid-19th-century efforts in to resolve failures of unique in rings of algebraic integers. developed the concept of ideal numbers in the as a means to restore unique in cyclotomic fields, particularly to prove cases of for regular primes. His 1844 paper introduced these "ideal complex numbers" to handle the decomposition of primes in extensions like the ring of integers of \mathbb{Q}(\zeta_p), where p is prime, addressing the non-unique observed in earlier attempts by Lamé. Building on Kummer's ideas, Richard Dedekind formalized the theory of ideals in the 1870s, shifting from abstract ideal numbers to concrete subsets of the ring of integers. In his 1871 supplement to Dirichlet's Vorlesungen über Zahlentheorie, Dedekind defined ideals as "systems" of multiples within quadratic fields, exemplified by the ring \mathbb{Z}[\sqrt{-5}], where the element 6 factors non-uniquely as $2 \cdot 3 = (1 + \sqrt{-5})(1 - \sqrt{-5}), but the corresponding ideals (2, 1 + \sqrt{-5}) and (3, 1 + \sqrt{-5}) are prime and distinct. This approach enabled unique factorization into ideals, laying groundwork for algebraic number theory. Concurrently, Leopold Kronecker pursued his Jugendtraum in the 1880s, envisioning abelian extensions of imaginary quadratic fields described via modular functions and class groups of rings of integers, linking ideal theory to class field theory. David Hilbert advanced these concepts into in the late 19th century, proving finiteness results that influenced ring structures. In his 1890 paper Über die Theorie der algebraischen Formen, Hilbert established the basis theorem, showing that the ring of invariants under linear group actions is finitely generated as a over the , resolving Gordan's algorithmic challenges with a non-constructive proof. His 1893 work Über die vollständige Theorie der algebraischen Formen extended these ideas to infinite systems of equations, emphasizing finite bases for syzygies in ideals. In the early 1900s, and later refined ideal decompositions, bridging and . Lasker's 1905 paper Zur Theorie der Moduln und Ideale introduced , proving that every ideal in a over a is a finite of primary ideals, providing an analogue to prime factorization. Noether generalized this in 1921 to Noetherian rings, establishing uniqueness up to radicals and solidifying the structure theory of ideals. These developments marked the transition toward abstract commutative algebra in the .

Key Figures and Milestones

played a pivotal role in the development of abstract during the 1920s, laying the groundwork for modern commutative algebra through her seminal 1921 paper "Idealtheorie in Ringbereichen," which established foundational concepts for ideals in ring domains. Her work emphasized abstract properties of rings, ideals, and modules over concrete examples, introducing the ascending chain condition that defines Noetherian rings—rings where every ascending chain of ideals stabilizes. also contributed the normalization lemma, which shows that any finitely generated can be viewed as integral over a subring, facilitating the study of algebraic varieties and integral extensions. By the mid-1920s, these innovations had transformed commutative algebra into a rigorous, axiomatic discipline. Wolfgang Krull advanced dimension theory in the 1930s, providing essential tools for understanding the structure of Noetherian rings through his principal ideal theorem, which bounds the height of minimal primes over a by one. This theorem marked a turning point in generalizing from polynomial rings to arbitrary Noetherian settings, enabling precise measurements of ring complexity via chains of prime ideals. Krull's contributions extended to broader intersection theorems, solidifying the algebraic framework for later geometric applications. Oscar Zariski, in the 1940s, integrated with by developing a purely algebraic topology on the , known as the , which endows the set of prime ideals with a structure suitable for studying varieties. During his time at from 1939 to 1940, Zariski applied modern to foundational questions in , introducing local rings and valuations to capture local properties of algebraic sets. His work on normality and birational equivalence bridged commutative structures with geometric intuition, influencing subsequent developments in scheme theory. Key milestones in the 1940s and 1950s included the Artin-Rees lemma, independently proved by and David Rees in the mid-1950s, which quantifies how powers of an ideal intersect with submodules, essential for completions and local cohomology. A special case had been known to Zariski earlier, but the general form enabled deeper analysis of filtrations in Noetherian modules. Irvin Sol Cohen established the structure theorem for complete Noetherian local rings in his 1942 dissertation under Oscar Zariski, published in 1946, showing that such rings are quotients of rings over fields or discrete valuation rings, providing a complete . The Bourbaki group, active from the 1930s onward, exerted significant influence in the 1950s through their series, particularly the volumes on algebra published in that decade, which standardized terminology and axiomatic treatments in commutative algebra. By emphasizing structuralist approaches and unifying concepts across , Bourbaki's rigorous expositions on rings, ideals, and modules shaped pedagogical and research standards, promoting a cohesive framework for the field.

Localization Techniques

Construction of Localizations

In commutative algebra, localization is a fundamental construction that allows one to invert a specified set of elements in a ring while preserving its . Given a R with identity and a multiplicative S \subseteq R (a subset containing 1 and closed under ), the localization S^{-1}R is formed as the set of equivalence classes of fractions r/s where r \in R and s \in S, with the r/s = r'/s' if there exists t \in S such that t(s'r - s r') = 0. Addition and are defined componentwise: (r/s) + (r'/s') = (r s' + r' s)/(s s') and (r/s)(r'/s') = (r r')/(s s'), making S^{-1}R a with identity $1/1. There is a \phi: R \to S^{-1}R given by r \mapsto r/1, whose consists of elements annihilated by some element of S. The localization S^{-1}R satisfies a universal property that characterizes it uniquely up to : for any f: R \to B such that f(s) is a in B for every s \in S, there exists a unique \overline{f}: S^{-1}R \to B making the diagram commute, i.e., f = \overline{f} \circ \phi. This property ensures that S^{-1}R is the "universal" ring extension of R in which all elements of S become invertible, facilitating the study of local behavior by inverting non-zero-divisors or elements outside ideals. Localization extends naturally to modules. For an R-module M, the localization S^{-1}M is the set of fractions m/s with m \in M, s \in S, under the analogous , and it becomes an S^{-1}R-module via the action (r/s) \cdot (m/t) = (r m)/(s t). There is a canonical S^{-1}M \cong S^{-1}R \otimes_R M, where the identifies (r/s) \otimes m = r/s \cdot (1 \otimes m), preserving exact sequences when S^{-1}R is flat over R. A prominent example is the field of fractions of an R, obtained by localizing at S = R \setminus \{0\}, yielding S^{-1}R = \mathrm{Frac}(R) where every non-zero element is invertible. Another key case arises when localizing at the complement of a : for a \mathfrak{p} \subseteq R, set S = R \setminus \mathfrak{p}; then S^{-1}R is a local ring with maximal ideal \mathfrak{p}(S^{-1}R) = \{ r/s \mid r \in \mathfrak{p}, s \in S \}, and the canonical map \phi identifies elements outside \mathfrak{p} as units.

Properties of Local Rings

A local ring is a commutative ring R that possesses exactly one maximal ideal, conventionally denoted \mathfrak{m}. The residue field of R is the quotient field k = R / \mathfrak{m}, which plays a central role in the study of modules over R. This structure arises naturally from localization techniques, where localizing a commutative ring at a prime ideal yields a ring with a unique maximal ideal consisting of elements whose numerators lie in that prime. One of the fundamental properties of local rings is captured by Nakayama's lemma, which provides a criterion for the vanishing of finitely generated modules. Specifically, if R is a local ring with maximal ideal \mathfrak{m} and M is a finitely generated R-module such that M = \mathfrak{m} M, then M = 0. An equivalent formulation states that if \{x_1, \dots, x_n\} is a set of elements in M whose images form a basis for the vector space M / \mathfrak{m} M over the residue field k, then these elements generate M as an R-module. This lemma is indispensable for proving generation and freeness results in module theory over local rings. For an R-module M, the support \operatorname{Supp}(M) is the closed subset V(\operatorname{Ann}_R(M)) of the spectrum \operatorname{Spec}(R), consisting of all prime ideals containing the annihilator ideal \operatorname{Ann}_R(M). In the context of a local ring R with maximal ideal \mathfrak{m}, the support is concentrated near \mathfrak{m}, and for finitely generated modules, this set determines key homological properties. If M is finitely generated, then \operatorname{Supp}(M) coincides precisely with V(\operatorname{Ann}_R(M)). Illustrative examples of local rings include discrete valuation rings (DVRs), which are one-dimensional regular local domains. A prototypical DVR is the localization \mathbb{Z}_{(p)} of the integers at the (p), for a p; here, the unique is p \mathbb{Z}_{(p)}, and the is \mathbb{F}_p. local rings generalize this structure; for instance, the power series ring k[[x_1, \dots, x_d]] over a k in d variables is a of dimension d, with generated by the x_i. These examples highlight how local rings encode local geometric or arithmetic information. Over an Artinian local ring R (which has finite length as an R- and thus is both left and right Artinian), every finitely generated M admits a finite , and the \ell_R(M) is the number of terms in any such series, invariant under isomorphism. This length function satisfies additivity: \ell_R(M \oplus N) = \ell_R(M) + \ell_R(N), and for the k, \ell_R(k) = 1. Artinian local rings often appear as quotients of regular s by ideals of finite .

Noetherian Rings

Definition and Basic Properties

A commutative ring R is called Noetherian if every ideal of R is finitely generated as an ideal. This condition is equivalent to the ascending chain condition (ACC) on ideals in R: for any ascending chain of ideals I_1 \subseteq I_2 \subseteq \cdots, there exists an integer n such that I_n = I_{n+1} = \cdots. It is also equivalent to every nonempty collection of ideals in R having a maximal element under inclusion. More generally, an R-module M (where R is commutative) is if every submodule of M is finitely generated. Equivalently, M satisfies the on submodules: every ascending chain of submodules stabilizes. A key basic property is that if R is a and M is a finitely generated R-module, then every submodule of M is also finitely generated. This follows from the fact that such an M admits a finite by cyclic submodules, each of which inherits the Noetherian property from R. Examples of Noetherian rings include the integers \mathbb{[Z](/page/Z)}, whose ideals are all principal and thus finitely generated, and polynomial rings k[x_1, \dots, x_n] in finitely many variables over a k. In contrast, the polynomial ring k[x_1, x_2, \dots ] in infinitely many variables over a k is not Noetherian, as it admits an infinite strictly ascending chain of ideals generated by the variables. To see why polynomial rings in finitely many variables satisfy the , consider monomial ideals. Dickson's states that in k[x_1, \dots, x_n], every ideal is finitely generated as a monomial ideal. The leading monomials (with respect to a ) of elements in any form a monomial ideal, so by Dickson's this set is finitely generated; selecting polynomials with these leading monomials yields a finite generating set for the original , implying the on ideals.

Hilbert Basis Theorem

The Hilbert basis theorem, proved by in 1890, asserts that if R is a , then the R in one indeterminate is also Noetherian. This means every ideal in R is finitely generated as an R-ideal. A standard proof proceeds by considering an arbitrary ideal I in R. For each nonnegative integer n, define b_n to be the ideal in R generated by the leading coefficients of all polynomials in I of degree at most n. Then b_0 \subseteq b_1 \subseteq \cdots is an ascending chain of ideals in R, which stabilizes because R is Noetherian: there exists d such that b_d = b_{d+1} = \cdots. Since b_d is finitely generated as an ideal in R, say by elements c_1, \dots, c_m, for each i choose a polynomial f_i \in I such that the leading coefficient of f_i is c_i and \deg f_i \leq d. Let \delta = \max \{ \deg f_i \} \leq d. The polynomials f_1, \dots, f_m generate I, because any g \in I of degree e \leq \delta has leading coefficient in b_e = b_\delta, so g minus a suitable R- of the f_i (shifted by powers of x if necessary) has lower degree or is zero; for e > \delta, the leading coefficient of g is in b_e = b_d, so again subtract a combination to reduce the degree below e, and induct on the degree to show g is in the ideal generated by the f_i. This argument implicitly relies on ideas akin to those in theory, where leading terms generate the ideal structure. The theorem generalizes immediately to multiple variables: if R is Noetherian, then so is the polynomial ring R[x_1, \dots, x_n] for any positive n, by iterated application of the univariate case. As a consequence, every in such a multivariate polynomial ring is finitely generated, implying that polynomial rings over Noetherian rings satisfy the ascending chain condition on . In particular, any generated by finitely many polynomials in R[x_1, \dots, x_n] admits a finite generating set, which establishes the finite basis property central to computational algebra. This result has significant applications in elimination theory, where the finite generation of ideals in rings allows for the effective of elimination ideals—subideals obtained by setting certain variables to zero—via methods like Gröbner bases, facilitating the of systems of equations.

Primary Decomposition

Primary Ideals

In commutative algebra, a proper ideal Q of a R with identity is called primary if, whenever ab \in Q for a, b \in R, either a \in Q or some power b^n \in Q with n \geq 1. This condition ensures that the R/Q has the property that every zero-divisor is . A key feature of primary ideals is that their \sqrt{Q} = \{ r \in R \mid r^k \in Q \text{ for some } k \geq 1 \} is a , denoted P, making Q a P-primary ideal. The notion of primary ideals connects closely to the associated primes of modules, which generalize the concept to R-modules. For an R-module M, the set of associated primes \mathrm{Ass}_R(M) consists of all prime ideals P such that P = \mathrm{Ann}_R(M/N) for some submodule N \subseteq M, where \mathrm{Ann}_R(M/N) = \{ r \in R \mid r \cdot (M/N) = 0 \}. Equivalently, P \in \mathrm{Ass}_R(M) if there exists an element m \in M such that P = \mathrm{Ann}_R(m), the annihilator of the cyclic submodule generated by m. In the case of an ideal I \subseteq R, the associated primes of I are precisely \mathrm{Ass}_R(R/I), linking primary decompositions directly to the primes that "support" the ideal. Examples of primary ideals abound in familiar rings. Every prime ideal is primary, as the condition simplifies to the prime property with n = 1. More interestingly, powers of prime ideals are primary: if P is prime, then P^n is P-primary for any n \geq 1. For instance, in the integers \mathbb{Z}, the ideal (p^n) generated by a prime p and positive integer n is primary with radical (p). In the polynomial ring k[x, y] over a field k, the ideal (x)^2 = (x^2) is primary with radical (x), since any product ab \in (x^2) implies that if a \notin (x^2), then a has no x-factor of order at least 2, forcing b to have arbitrarily high powers absorbable into the ideal via nilpotency in the quotient. Similarly, \mathfrak{m}^n for a maximal ideal \mathfrak{m} provides a primary ideal whose radical is \mathfrak{m}. A fundamental property in Noetherian rings—that is, rings where every ascending chain of ideals stabilizes—states that every proper ideal admits a as a finite of primary ideals. This underpins the structure of ideals in such rings, allowing complex ideals to be broken down into "irreducible" primary components whose radicals reveal the underlying prime structure.

Unique Factorization into Primaries

In a R, the Lasker–Noether theorem states that every ideal I \subseteq R can be expressed as a finite intersection of primary ideals, I = \bigcap_{i=1}^n Q_i, where each Q_i is primary. This result, originally established by , generalizes earlier work by on decompositions in polynomial rings. The finiteness of the decomposition follows directly from the ascending chain condition on ideals in . The is not necessarily , but it admits an irredundant form, meaning that for each i, Q_i \not\supseteq \bigcap_{j \neq i} Q_j. In any two irredundant primary decompositions of I, the ideals P_i = \sqrt{Q_i} coincide as sets; these are the primes in the of the R- R/I. Moreover, the primary components corresponding to minimal associated primes (those not properly containing any other associated prime) are : for a minimal P, the corresponding Q satisfies Q = I R_P \cap R, where R_P is the localization of R at P. For non-minimal () associated primes, the individual primary components may vary across decompositions, but their intersection for each fixed embedded prime is . The primary ideals in the decomposition can be classified as isolated or based on their radicals. Isolated components correspond to minimal associated primes, while embedded components arise from non-minimal ones, where P_i \subsetneq P_j for some j. This distinction highlights how the decomposition captures both the "essential" structure of the ideal (via isolated parts) and additional "overlaps" (via embedded parts). In the formula I = \bigcap Q_i with distinct \sqrt{Q_i} = P_i, the P_i form the complete set of associated primes. A concrete illustration occurs in the k[x,y] over a k, with the I = (x^2 y, x y^2, y^3). This equals y (x,y)^2, and its irredundant is I = (y) \cap (x,y)^3, where (y) is primary to the minimal prime (y) (isolated component) and (x,y)^3 is primary to the embedded prime (x,y), since (y) \subsetneq (x,y). To verify, note that the generators of I lie in both ideals, and any element in the intersection must vanish to order at least 3 along the line x=y=0 while also being divisible by y, matching I; conversely, higher terms like x^3 lie in (x,y)^3 but not in (y).

Dimension and Integral Extensions

Krull Dimension

The Krull dimension of a commutative ring R, named after Wolfgang Krull, is defined as the supremum of the lengths of all strictly ascending chains of s in R. A chain \mathfrak{p}_0 \subsetneq \mathfrak{p}_1 \subsetneq \dots \subsetneq \mathfrak{p}_n of s has length n, where the inclusions are strict, and the dimension \dim R is the largest such n if it exists, or infinite otherwise. This definition provides an algebraic measure of the "size" of the ring in terms of its structure, and for Noetherian rings, the dimension is finite. For an R-module M, the Krull dimension \dim M is defined as the Krull dimension of the quotient ring R / \Ann_R(M), where \Ann_R(M) is the annihilator ideal of M. This extends the notion to modules by associating to M the dimension of the ring acting effectively on it, capturing the complexity of the support of M in \Spec R. Key properties of the Krull dimension include the behavior under polynomial extensions: for a Noetherian ring R, \dim R = \dim R + 1. Rings in which all maximal chains of prime ideals between a given minimal prime and a given maximal prime have the same length are called catenary; many important classes of rings, such as Cohen-Macaulay rings and regular local rings, are catenary. Krull's principal ideal theorem states that in a Noetherian ring R, if \mathfrak{p} is a prime ideal minimal over a principal ideal (a), then the height \ht \mathfrak{p} \leq 1. The height \ht \mathfrak{p} of a prime ideal \mathfrak{p} is itself the Krull dimension of the localization R_\mathfrak{p}. Examples illustrate these concepts clearly. The \mathbb{Z} has 1, as every chain of prime ideals is of the form (0) \subsetneq (p) for a p. The k[x,y] in two variables over a k has dimension 2, exemplified by the chain (0) \subsetneq (x) \subsetneq (x,y). A local , such as the power series ring k[] over a k, has dimension 1, with the unique nonzero (x) of height 1.

Integral Closure and Normal Rings

An element b in a B containing a A is said to be over A if there exists a f(T) = T^n + a_{n-1} T^{n-1} + \cdots + a_0 with coefficients a_i \in A such that f(b) = 0. The set of all elements in B over A forms a of B, called the integral closure of A in B and denoted \overline{A}. An extension A \subseteq B of commutative rings is integral if every element of B is over A. For an R with quotient field K, the integral closure \overline{R} is taken in K, and R is called (or integrally closed) if R = \overline{R}. Integrality is transitive: if A \subseteq B \subseteq C are commutative rings with B over A and C over B, then C is over A. A fundamental result is the lying-over theorem: if A \subseteq B is an integral extension of commutative rings, then for every prime ideal \mathfrak{p} of A, there exists a prime ideal \mathfrak{q} of B such that \mathfrak{q} \cap A = \mathfrak{p}. In such extensions, prime ideals of B lying over the same prime ideal of A are incomparable, meaning no one contains another. The going-up theorem states that if \mathfrak{p} \subseteq \mathfrak{p}' are prime ideals in A and \mathfrak{q} is a prime ideal in B lying over \mathfrak{p}, then there exists a prime ideal \mathfrak{q}' in B such that \mathfrak{q} \subseteq \mathfrak{q}' and \mathfrak{q}' \cap A = \mathfrak{p}'. The going-down theorem holds under additional hypotheses: if A and B are integral domains with A normal and B integral over A, then if \mathfrak{p}_1 \subseteq \mathfrak{p}_2 are prime ideals in A and \mathfrak{q}_2 is a prime ideal in B lying over \mathfrak{p}_2, there exists a prime ideal \mathfrak{q}_1 in B such that \mathfrak{q}_1 \subseteq \mathfrak{q}_2 and \mathfrak{q}_1 \cap A = \mathfrak{p}_1. These properties ensure that integral extensions preserve the structure of prime ideal chains in a controlled manner. A classic example is the integers \mathbb{Z} in the rationals \mathbb{Q}: every element of \mathbb{Q} integral over \mathbb{Z} is already in \mathbb{Z}, so \mathbb{Z} is normal. Another example arises in algebraic geometry: consider the ring A = k[x,y]/(y^2 - x^3) over a field k, the coordinate ring of the cuspidal cubic curve. This ring is not normal, but its integral closure in the function field is k, where x = t^2 and y = t^3, yielding the normalization map that resolves the cusp singularity. For integral extensions A \subseteq B of commutative rings, the Krull dimensions satisfy \dim B = \dim A. This equality follows from the lying-over and going-up theorems, which allow chains of prime ideals in A to lift to chains of equal length in B, and vice versa.

Completions

Adic Completions

In commutative algebra, the I-adic topology on a ring R with respect to a proper ideal I is defined by taking the powers I^n as a fundamental system of neighborhoods of zero, making the open sets those containing some I^n. This topology induces a metric where the distance between elements is determined by the highest power of I separating them, turning R into a topological ring. The I-adic completion \hat{R} of R is constructed as the inverse limit \hat{R} = \lim_{\leftarrow n} R / I^n, where the transition maps are the natural projections R / I^{n+1} \to R / I^n. Equivalently, elements of \hat{R} can be represented as Cauchy sequences (a_n) in R with respect to the , modulo the identifying sequences that converge to zero (null sequences), where a sequence is Cauchy if for every m, there exists N such that a_{n} - a_{k} \in I^m for all n, k \geq N. The ring operations on \hat{R} are defined componentwise, making it a complete topological ring with respect to the induced topology. A key feature of the completion is its : for any \phi: \to S where S is complete with respect to the J-adic topology for some ideal J \subseteq S with \phi(I) \subseteq J, there exists a unique continuous extension \hat{\phi}: \hat{R} \to S such that \hat{\phi} \circ \pi = \phi, where \pi: \to \hat{R} is the . When is Noetherian, \hat{R} is flat as an R-module, a consequence of the Artin-Rees lemma which ensures that completion preserves exact sequences under suitable conditions. Furthermore, for a complete Noetherian local ring (, \mathfrak{m}) of characteristic p or mixed characteristic, the Cohen structure theorem states that is a quotient of a power series ring over a coefficient field or a discrete valuation ring, providing a regular model for such rings. Prominent examples include the p-adic integers \mathbb{Z}p, which form the (p)-adic of \mathbb{Z}, consisting of formal series \sum{i=0}^\infty a_i p^i with a_i \in {0, \dots, p-1}. Another is the ring k[] over a k, which is the (x)-adic of the localized k_{(x)}.

Applications to Power Series Rings

The ring R[] over a R consists of all infinite formal sums \sum_{n=0}^\infty a_n x^n with coefficients a_n \in R. Addition is defined componentwise, while multiplication is given by the formula: if f = \sum_{n=0}^\infty a_n x^n and g = \sum_{m=0}^\infty b_m x^m, then f g = \sum_{k=0}^\infty c_k x^k, where c_k = \sum_{i=0}^k a_i b_{k-i}. This ring arises as the completion of the polynomial ring R localized at the prime ideal (x), denoted R[] \cong \widehat{(R)_{(x)}}, with respect to the maximal ideal (x) R_{(x)}. As a specific instance of adic completion, it equips the polynomial ring with a where sequences of polynomials converging coefficientwise extend to infinite series. If R is Noetherian, then R[] is also Noetherian. A key property of R[] is the structure of its ideals: when R is a field k, the ideal (x) (comprising series with zero constant term) is the unique maximal ideal, making k[] a local ring. More generally, elements of R[] are units if and only if their constant term is a unit in R. The Weierstrass preparation theorem provides a factorization tool in this setting: for a power series f \in R[] over a complete local ring R with maximal ideal \mathfrak{m}, if f is regular of order d (i.e., f \equiv u x^d \pmod{\mathfrak{m}} for a unit u \in R), then f = u(x) \cdot \pi(x), where u(x) is a unit in R[] and \pi(x) is a distinguished polynomial of degree d (Weierstrass polynomial). This theorem facilitates division algorithms and uniqueness in power series expansions. Examples illustrate these properties effectively. Over a k, the k[] is a complete with uniformizer x and valuation given by the lowest nonzero degree. rings also play a central role in deformation theory, where they parametrize deformations of algebraic objects, such as modules or schemes, over artinian base rings like truncated .

Connections to Algebraic Geometry

Spectrum of a Ring

The prime of a R, denoted \operatorname{Spec}(R), is the set consisting of all prime of R, equipped with the . In this topology, the closed subsets are precisely the sets of the form V(I) = \{ \mathfrak{p} \in \operatorname{Spec}(R) \mid I \subseteq \mathfrak{p} \}, where I is any of R. The open sets admit a basis given by the principal open subsets D(f) = \{ \mathfrak{p} \in \operatorname{Spec}(R) \mid f \notin \mathfrak{p} \} for elements f \in R; moreover, each such D(f) is homeomorphic to \operatorname{Spec}(R_f), the of the localization of R at the multiplicative set generated by f. This topological structure captures the "points" of the ring in a way that reflects its ideal-theoretic properties, with generic points corresponding to prime ideals contained in many others. To elevate \operatorname{Spec}(R) from a mere to a suitable for , one defines the structure sheaf \mathcal{O}_{\operatorname{Spec}(R)} of rings on it. On each basic open D(f), the sections are \Gamma(D(f), \mathcal{O}_{\operatorname{Spec}(R)}) = R_f, with restriction maps to the intersection D(fg) = D(f) \cap D(g) given by the natural localization R_f \to R_{fg}. This presheaf extends uniquely to a sheaf, and the stalks at points \mathfrak{p} \in \operatorname{Spec}(R) are the local rings R_{\mathfrak{p}}. The resulting locally (\operatorname{Spec}(R), \mathcal{O}_{\operatorname{Spec}(R)}) is an affine , serving as the fundamental building block for scheme theory. The Zariski topology on \operatorname{Spec}(R) endows it with several key topological properties: it is always (every irreducible closed subset has a unique ) and quasi-compact (every open cover has a finite subcover). However, it is generally not Hausdorff, as specializations between s prevent separation of distinct points by disjoint opens; specifically, \operatorname{Spec}(R) is Hausdorff if and only if every of R is maximal (equivalently, R is zero-dimensional). These features make \operatorname{Spec}(R) a spectral space in the sense of Hochster, precisely the class of spaces arising as prime spectra. Illustrative examples highlight the geometry encoded in the spectrum. For R = \mathbb{Z}, \operatorname{Spec}(\mathbb{Z}) comprises the zero ideal (0) (the , dense in the space) and the maximal ideals (p) for prime numbers p (closed points); the V((n)) consists of (0) and the (p) dividing n > 0. For R = k with k an , \operatorname{Spec}(k) models the affine line \mathbb{A}^1_k, where points correspond to the zero ideal (generic) and principal ideals (x - a) for a \in k (closed points), or more generally (f) for irreducible f. This topological framework underpins the connection to affine schemes and varieties in .

Affine Schemes

An affine scheme is defined as a pair (\operatorname{Spec} R, \mathcal{O}), where R is a , \operatorname{Spec} R is the spectrum of R equipped with the , and \mathcal{O} is the structure sheaf such that for any basic D(f) \subseteq \operatorname{Spec} R (where f \in R), the sections \mathcal{O}(D(f)) = R_f, the localization of R at the multiplicative set generated by f. This structure sheaf assigns to each a ring of "regular functions" on that set, making the affine scheme a locally that bridges commutative algebra to geometric objects. The points of an affine scheme \operatorname{Spec} R over an algebraically closed field k correspond bijectively to ring homomorphisms R \to \overline{k}, where \overline{k} is an algebraically closed field extension (often taken as k itself), via the evaluation at maximal ideals. A foundational result establishing this correspondence is Hilbert's Nullstellensatz, which states that for an algebraically closed field k, the maximal ideals of the polynomial ring k[x_1, \dots, x_n] are precisely the ideals of the form (x_1 - a_1, \dots, x_n - a_n) for points (a_1, \dots, a_n) \in k^n. This theorem identifies the closed points of the affine scheme \operatorname{Spec} k[x_1, \dots, x_n] with the classical affine space k^n, providing a geometric interpretation of maximal ideals. Regular functions on an affine \operatorname{Spec} R are elements of the sections \mathcal{O}(\operatorname{Spec} R) = R, which consist of polynomials or more generally elements that are locally ratios of such on basic opens. For a V \subseteq \operatorname{Spec} R defined as the zero set of an ideal I, the vanishing ideal I(V) is the of the ideal generated by elements of R that vanish on V, capturing the polynomials defining the subvariety scheme-theoretically. By the weak form of the Nullstellensatz, if k is algebraically closed, then I(V(I)) = \sqrt{I} for any ideal I in k[x_1, \dots, x_n], ensuring a between radical ideals and affine varieties. Affine schemes serve as building blocks for more general schemes; for instance, the projective scheme \operatorname{Proj} S for a S is covered by affine open subschemes D_+(f) isomorphic to \operatorname{Spec} S_{(f)}, where S_{(f)} is the degree-zero part of the localization at the homogeneous element f. General schemes are obtained by gluing affine schemes along isomorphisms of their intersections, with the structure sheaf defined via the sheaf property on the affine cover. This gluing construction allows projective spaces, such as \mathbb{P}^n_k = \operatorname{Proj} k[x_0, \dots, x_n], to be realized as unions of affine charts D_+(x_i) \cong \mathbb{A}^n_k.

Applications to Number Theory

Dedekind Domains

A is defined as an of one, in which every nonzero is . This structure ensures that the domain is a one-dimensional that is , meaning its in its coincides with itself. Equivalently, a can be characterized as a where every nonzero is and the localization at every is a . A key property of Dedekind domains is the unique factorization of ideals: every nonzero proper ideal factors uniquely as a product of prime ideals, up to ordering and units. More precisely, for any nonzero ideal I in a Dedekind domain R, there exist unique prime ideals \mathfrak{p}_1, \dots, \mathfrak{p}_n and positive integers e_1, \dots, e_n such that I = \mathfrak{p}_1^{e_1} \cdots \mathfrak{p}_n^{e_n}. This factorization theorem extends the unique factorization of elements in principal ideal domains but applies to ideals rather than elements, making Dedekind domains central to algebraic number theory. Examples of Dedekind domains include the ring of integers \mathbb{Z}, which is a principal ideal domain and thus satisfies the definition. More generally, the ring of integers in any number field is a Dedekind domain, as it is the integral closure of \mathbb{Z} in the field and has dimension one. In contrast, \mathbb{Z}[\sqrt{-3}] is not a Dedekind domain because it fails to be integrally closed; the element \frac{1 + \sqrt{-3}}{2} satisfies the monic polynomial x^2 - x + 1 = 0 over \mathbb{Q}(\sqrt{-3}) but lies outside \mathbb{Z}[\sqrt{-3}]. The of a R is the of the group of fractional ideals by the of principal fractional ideals, measuring the deviation from unique element . For the in a number field, this group is finite, as established using .

Class Groups and Ideal Factorization

In Dedekind domains, every nonzero fractional ideal is invertible, and the Picard group \Pic(R) is defined as the group of isomorphism classes of invertible ideals under multiplication, modulo the subgroup of principal fractional ideals. For a Dedekind domain R, \Pic(R) coincides with the ideal class group \Cl(R), which measures the failure of unique factorization into prime ideals up to principal ideals. This group is abelian and finitely generated when R is the ring of integers of a number field. The finiteness of the class number h(R) = |\Cl(R)| for R = \mathcal{O}_K the of a number field K of n = [K:\mathbb{Q}] follows from Minkowski's theorem. Every ideal class contains an integral I \subseteq \mathcal{O}_K with N(I) \leq m_K, where the Minkowski constant is m_K = \frac{n!}{n^n} \left( \frac{4}{\pi} \right)^{r_2} \sqrt{|\Delta_K|}, with r_2 the number of complex conjugate pairs of embeddings and \Delta_K the . Since there are only finitely many prime ideals of norm at most m_K, the class group is finite. This bound enables explicit computation of \Cl(\mathcal{O}_K) by finding representatives and relations among ideals of small norm. In extensions of Dedekind domains, the decomposition of prime ideals illustrates how ideals factor. For a quadratic extension K = k(\sqrt{d}) with k = \mathbb{Q} and d > 0 or d < 0, an odd prime p of k not dividing the discriminant \Delta_K = 4d or d (depending on congruence) decomposes as follows: it splits into two distinct primes if \left( \frac{\Delta_K}{p} \right) = 1, remains inert (prime in \mathcal{O}_K) if \left( \frac{\Delta_K}{p} \right) = -1, or ramifies (squares to a prime) if \left( \frac{\Delta_K}{p} \right) = 0. Primes dividing $2or\Delta_K require separate analysis but follow similar inert/split/ramified behavior based on quadratic residues modulo &#36;4 or $8. This factorization governs the structure of \Cl(\mathcal{O}_K)$ via the Artin symbol. Examples of class numbers in quadratic fields \mathbb{Q}(\sqrt{d}) highlight these concepts. For imaginary quadratic fields (d < 0), \mathbb{Q}(\sqrt{-1}) and \mathbb{Q}(\sqrt{-3}) have class number $1, while \mathbb{Q}(\sqrt{-5}) has class number &#36;2, computed by verifying no ideals of norm up to the Minkowski bound m_K \approx 3.16 generate non-principal classes. For real quadratic fields (d > 0), \mathbb{Q}(\sqrt{2}), \mathbb{Q}(\sqrt{5}), and \mathbb{Q}(\sqrt{13}) have class number $1, but \mathbb{Q}(\sqrt{10}) has class number &#36;2. Computations often use the expansion of \sqrt{d} to find the fundamental unit \varepsilon, which aids in reducing ideals; the period length of the expansion relates to the narrow class group, and reduced binary quadratic forms of \Delta_K enumerate the classes, with the class number equaling their count for principal . For extensions R \to S of Dedekind domains where S is the closure in a finite unramified abelian extension of \mathrm{Frac}(R), the class number satisfies h(R) divides h(S), with the quotient termed the relative class number. This follows from the Artin reciprocity map in , where the embeds into the idele class group, inducing a surjection \Cl(S) \twoheadrightarrow \Cl(R) whose kernel size yields the relative factor. In the Hilbert class field H of K = \mathrm{Frac}(R), unramified over K with [\Gal(H/K)] = h(R), this divisibility holds explicitly.