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Commutative ring

A commutative ring is an algebraic structure consisting of a set R equipped with two binary operations, addition and multiplication, such that (R, +) forms an abelian group, multiplication is associative and commutative (i.e., ab = ba for all a, b \in R), and multiplication distributes over addition (i.e., a(b + c) = ab + ac and (a + b)c = ac + bc for all a, b, c \in R). Many treatments of commutative rings also require the presence of a multiplicative identity element, denoted 1, satisfying $1 \cdot a = a \cdot 1 = a for all a \in R. Commutative rings generalize familiar structures such as the integers \mathbb{Z}, the rational numbers \mathbb{Q}, the real numbers \mathbb{R}, and polynomial rings like \mathbb{Z}, providing a framework for studying arithmetic properties in a unified way. Key subconcepts include ideals, which are subsets closed under addition and absorption by ring elements, enabling the construction of quotient rings analogous to modular arithmetic. Special cases of commutative rings encompass integral domains, where there are no zero-divisors (nonzero elements whose product is zero), and fields, where every nonzero element has a multiplicative inverse. Commutative ring theory forms the core of commutative algebra, a branch of abstract algebra that explores rings and their modules, with profound applications across mathematics. In number theory, it underpins the study of algebraic integers and Dedekind domains for analyzing prime factorization in number fields. Algebraic geometry relies on commutative rings to model varieties through their coordinate rings, linking geometric objects to algebraic equations. Further applications appear in singularity theory for understanding local properties of algebraic varieties, homological algebra for computing invariants, and even cryptography via elliptic curves over finite fields. Commutative rings also arise naturally in algebraic topology, graph theory, and combinatorics through symmetry studies and invariant computations.

Foundations

Definition

A is a that generalizes the properties of integers under and , serving as a foundational object in . Formally, a commutative ring R is a set equipped with two operations, + and \cdot (often denoted by juxtaposition ab), satisfying the following axioms: (R, +) forms an with $0, meaning is associative and commutative, every element has an , and $0 + a = a for all a \in R; is associative, i.e., (ab)c = a(bc) for all a, b, c \in R; distributes over from both sides, i.e., a(b + c) = ab + ac and (b + c)a = ba + ca for all a, b, c \in R; and is commutative, i.e., ab = ba for all a, b \in R. Additionally, commutative rings are typically required to have a multiplicative $1 \in R such that $1 \cdot a = a \cdot 1 = a for all a \in R, distinguishing them from rngs, which are similar structures lacking this . The notation for a commutative ring is often (R, +, \cdot, 0, 1), and subrings are subsets inheriting these operations. The , where R = \{0\} and $0 = 1, satisfies the axioms but is often excluded in certain contexts due to its degenerate nature. The characteristic of a commutative ring R, denoted \operatorname{char}(R), is the smallest positive integer n such that n \cdot 1 = 0 (where n \cdot 1 = 1 + \cdots + 1 with n summands), or $0 if no such finite n exists. This measures the "size" of the ring's additive structure relative to the integers.

Basic Examples

The ring of integers \mathbb{Z} under the usual addition and multiplication operations serves as the prototypical example of a commutative ring with unity, where the additive identity is 0 and the multiplicative identity is 1. In \mathbb{Z}, every element has an additive inverse, and multiplication is commutative and distributive over addition, satisfying all ring axioms without zero divisors other than 0 itself. Fields such as the rational numbers \mathbb{Q}, real numbers \mathbb{R}, and complex numbers \mathbb{C} are commutative rings where every nonzero element has a multiplicative inverse, making them integral domains and in fact fields. For instance, in \mathbb{Q}, and multiplication follow the standard field operations, with 1 as the unity and no zero divisors. The k over a k, consisting of with coefficients in k and operations of and , forms a commutative ring with (the constant polynomial 1). is term-wise, and uses the , ensuring commutativity since k is commutative; for example, over \mathbb{R}, \mathbb{R} includes elements like x^2 + 3x - 2. Quotient rings like \mathbb{Z}/6\mathbb{Z}, the integers modulo 6 with component-wise and modulo 6, provide examples of commutative rings with zero divisors. Here, are equivalence classes {{grok:render&&&type=render_inline_citation&&&citation_id=0&&&citation_type=wikipedia}} through {{grok:render&&&type=render_inline_citation&&&citation_id=5&&&citation_type=wikipedia}}, and zero divisors include {{grok:render&&&type=render_inline_citation&&&citation_id=2&&&citation_type=wikipedia}} and {{grok:render&&&type=render_inline_citation&&&citation_id=3&&&citation_type=wikipedia}} since {{grok:render&&&type=render_inline_citation&&&citation_id=2&&&citation_type=wikipedia}} \cdot {{grok:render&&&type=render_inline_citation&&&citation_id=3&&&citation_type=wikipedia}} = {{grok:render&&&type=render_inline_citation&&&citation_id=0&&&citation_type=wikipedia}} but neither is {{grok:render&&&type=render_inline_citation&&&citation_id=0&&&citation_type=wikipedia}}. The R \times S of two commutative rings R and S, with component-wise and , is itself a commutative ring with unity (1_R, 1_S). For example, \mathbb{Z} \times \mathbb{Z} has like (a, b) where is (a, b) + (c, d) = (a+c, b+d) and is (a, b) \cdot (c, d) = (ac, bd), preserving commutativity from R and S. While full matrix rings M_n(k) over a field k for n \geq 2 are generally non-commutative, the subring of diagonal matrices—those with off-diagonal entries zero—forms a commutative ring isomorphic to k^n under component-wise operations. For instance, in M_2(\mathbb{R}), diagonal matrices like \begin{pmatrix} a & 0 \\ 0 & b \end{pmatrix} multiply commutatively as \begin{pmatrix} a & 0 \\ 0 & b \end{pmatrix} \begin{pmatrix} c & 0 \\ 0 & d \end{pmatrix} = \begin{pmatrix} ac & 0 \\ 0 & bd \end{pmatrix}.

Algebraic Structures

Ideals

In a commutative ring R, an is defined as an additive I of R such that r \cdot i \in I for all r \in R and i \in I. This property ensures that ideals are substructures closed under by arbitrary ring elements. In noncommutative rings, one must distinguish between left ideals (where r \cdot i \in I), right ideals (where i \cdot r \in I), and two-sided ideals (satisfying both), but in the commutative case, these distinctions vanish since multiplication is symmetric, and all ideals coincide with two-sided ideals. The principal ideal generated by a single element a \in R is the set (a) = \{ r \cdot a \mid r \in R \}, which consists of all multiples of a by ring elements. More generally, the ideal generated by a subset S \subseteq R is the smallest ideal containing S, formed by all finite sums \sum r_k s_k where r_k \in R and s_k \in S. If S is finite, say S = \{a_1, \dots, a_n\}, then the ideal is finitely generated and denoted (a_1, \dots, a_n). Ideals support natural operations that preserve the ideal structure. The sum of two ideals I and J is I + J = \{ i + j \mid i \in I, j \in J \}, the smallest ideal containing both I and J. The intersection I \cap J is itself an ideal, consisting of elements common to both. The product I \cdot J is the ideal generated by all products i \cdot j with i \in I and j \in J, explicitly the set of all finite sums \sum i_k j_k. For example, in the ring of integers \mathbb{Z}, every nonzero ideal is principal of the form (n) = n\mathbb{Z} for some positive integer n. In this case, the sum of principal ideals satisfies (a) + (b) = (d), where d = \gcd(a, b).

Quotient Rings

In a commutative ring R with an ideal I, the quotient ring R/I is constructed as the set of cosets \{a + I \mid a \in R\}, where addition and multiplication are defined by (a + I) + (b + I) = (a + b) + I and (a + I)(b + I) = ab + I. These operations inherit the commutativity and distributivity from R, and I serves as the zero element in R/I, making it a commutative ring with identity \overline{1} = 1 + I. The natural projection \pi: R \to R/I given by \pi(a) = a + I is a surjective ring homomorphism with kernel I. The R/I satisfies a : for any commutative ring S and \phi: R \to S such that I \subseteq \ker \phi, there exists a unique \overline{\phi}: R/I \to S such that \phi = \overline{\phi} \circ \pi. This induces a natural of sets \operatorname{Hom}(R/I, S) \cong \{\phi: R \to S \mid I \subseteq \ker \phi\}, where the sends \overline{\phi} to \phi = \overline{\phi} \circ \pi. This property characterizes R/I as the "universal" ring obtained by forcing elements of I to zero. The first theorem for rings states that if \phi: R \to S is a between commutative rings, then R / \ker \phi \cong \operatorname{[im](/page/IM)} \phi as rings, where the is induced by the map a + \ker \phi \mapsto \phi(a). This theorem shows that every homomorphic image of R is isomorphic to a quotient of R by the kernel of the homomorphism. A key result involving quotients is the : if I and J are ideals in R such that I + J = R, then R / (I \cap J) \cong R/I \times R/J as rings, via the map a + (I \cap J) \mapsto (a + I, a + J), which is an . More generally, for pairwise coprime ideals I_1, \dots, I_n (meaning I_i + I_j = R for i \neq j), the natural map R \to \prod_{i=1}^n R/I_i has \bigcap_{i=1}^n I_i and is surjective, yielding R / \bigcap I_i \cong \prod R/I_i. Representative examples illustrate these constructions. The ring \mathbb{Z}/n\mathbb{Z} is the quotient of the integers \mathbb{Z} by the ideal n\mathbb{Z}, consisting of residue classes modulo n with componentwise and . For a k and f(x) \in k, the k / (f(x)) forms an , often a of k of degree \deg f.

Integral Domains

An integral domain is a commutative ring with multiplicative identity $1 \neq 0 in which there are no zero divisors, meaning that if ab = 0 for elements a, b in the ring, then either a = 0 or b = 0. This condition ensures cancellativity under multiplication: if ab = ac and a \neq 0, then b = c. In an R, a is a nonzero u \in R that has a u^{-1} \in R such that uu^{-1} = 1. Two nonzero elements a, b \in R are associates if there exists a u \in R such that a = ub; associates share the same divisibility properties and play a key role in uniqueness up to units. Euclidean domains form an important subclass of domains, equipped with a Euclidean function d: R \setminus \{0\} \to \mathbb{N} \cup \{0\} satisfying d(a) \leq d(ab) for all nonzero a, b \in R and a : for any a, b \in R with b \neq 0, there exist q, r \in R such that a = bq + r where either r = 0 or d(r) < d(b). This structure enables algorithms like the Euclidean algorithm for computing greatest common divisors. Examples include the integers \mathbb{Z} with d(n) = |n| and polynomial rings k over a field k with d(f) = \deg f. A principal ideal domain (PID) is an integral domain in which every ideal is principal, generated by a single element. Every Euclidean domain is a PID, but the converse does not hold; for instance, \mathbb{Z} and k (where k is a field) are PIDs. Unique factorization domains (UFDs) are integral domains where every nonzero non-unit element factors uniquely into irreducible elements, up to order and associates. Every PID is a UFD, as irreducibles in a PID coincide with primes, ensuring unique factorization via Noetherian properties and prime ideal structure. However, not every UFD is a PID, such as polynomial rings in multiple variables over a field. In a PID, the greatest common divisor d = \gcd(a, b) of two elements a, b generates the ideal they form, satisfying (a, b) = (d). (a, b) = \{ ra + sb \mid r, s \in R \} = (d)

Localization

Localizations

In commutative algebra, the localization of a commutative ring R at a multiplicative subset S \subseteq R (a subset containing $1and closed under multiplication) is constructed as the ringS^{-1}Rof fractionsr/swithr \in Rands \in S, where two fractions r/sandr'/s'are identified if there existst \in Ssuch thatt(s'r - sr') = 0. Addition and multiplication in S^{-1}Rare defined componentwise as(r/s) + (r'/s') = (rs' + r's)/(ss')and(r/s)(r'/s') = (rr')/(ss'), making S^{-1}Ra commutative ring with identity1/1$. The localization S^{-1}R satisfies a universal property: for any commutative ring T, the ring homomorphisms S^{-1}R \to T are in natural bijection with the ring homomorphisms \phi: R \to T such that \phi(s) is a unit in T for every s \in S. This property characterizes S^{-1}R up to unique isomorphism and justifies its role in inverting elements of S while preserving the structure of R. There is a canonical ring homomorphism \eta: R \to S^{-1}R given by \eta(r) = r/1, which is universal among maps from R to rings where elements of S become units. The kernel of \eta consists precisely of those elements r \in R annihilated by some element of S, i.e., \ker \eta = \{ r \in R \mid \exists s \in S \text{ with } sr = 0 \}. Examples of localizations include the case where S = \{1, a, a^2, \dots \} for a fixed element a \in R, which inverts all powers of a and is useful for studying behavior near the (a). Another common example arises when S is the complement of a prime \mathfrak{p} \subset R, localizing at S = R \setminus \mathfrak{p} to focus on elements outside \mathfrak{p}. For any ideal I \subseteq R, the localization S^{-1}I (defined as the image of I under \eta, i.e., fractions with numerators in I) forms an ideal of S^{-1}R, and it equals the product ideal S^{-1}R \cdot (S^{-1}I). This extension property ensures that ideals behave compatibly under localization, allowing the study of global ring properties through local ones.

Fraction Fields

For an integral domain R, the field of fractions, often denoted \operatorname{Frac}(R) or Q(R), is the localization of R at the multiplicative set S = R \setminus \{0\}. It can be constructed as the set of equivalence classes of ordered pairs (a, b) where a \in R, b \in R \setminus \{0\}, under the relation (a, b) \sim (c, d) if and only if ad = bc. This equivalence relation ensures that the construction is well-defined and models the intuitive notion of fractions in R. The field operations are defined componentwise: addition by (a, b) + (c, d) = (ad + bc, bd) and multiplication by (a, b) \cdot (c, d) = (ac, bd). These operations make \operatorname{Frac}(R) into a field, with the additive identity (0, 1) and multiplicative identity (1, 1), and every nonzero element having an inverse. There is a natural ring homomorphism \iota: R \to \operatorname{Frac}(R) given by a \mapsto (a, 1), which is injective because R has no zero divisors, thus embedding R as a subring of \operatorname{Frac}(R). Consequently, every integral domain embeds into a field in this manner, and \operatorname{Frac}(R) is unique up to isomorphism over R. Classic examples include the field of rational numbers \mathbb{Q} as \operatorname{Frac}(\mathbb{Z}), where elements are fractions a/b with a, b \in \mathbb{Z} and b \neq 0, reduced by the equivalence. Similarly, for a field k, the rational function field k(x) is \operatorname{Frac}(k), consisting of quotients f(x)/g(x) with g(x) \neq 0. In the context of function fields, the transcendence degree provides a measure of the "dimension" of the extension \operatorname{Frac}(R)/k when R is a domain over a field k. For the polynomial ring k[x_1, \dots, x_n], the transcendence degree of k(x_1, \dots, x_n) over k is n, reflecting the algebraic independence of the indeterminates. This degree is the size of a transcendence basis, a maximal algebraically independent set over k.

Modules

Definition and Properties

In the context of commutative rings, a module provides a structure that generalizes vector spaces, where the scalars come from a field, to the setting of a commutative ring R with identity. An R-module M is an abelian group (M, +) equipped with a scalar multiplication operation \cdot: R \times M \to M satisfying the following axioms for all r, s \in R and m, n \in M:
  • (r + s) \cdot m = r \cdot m + s \cdot m (distributivity over ring addition),
  • r \cdot (m + n) = r \cdot m + r \cdot n (distributivity over module addition),
  • (rs) \cdot m = r \cdot (s \cdot m) (associativity),
  • $1 \cdot m = m (identity preservation).
A subset N \subseteq M is a submodule if it forms an abelian subgroup under addition and is closed under scalar multiplication by elements of R, inheriting the operations from M. Given a submodule N of M, the quotient module M/N is defined as the set of cosets \{m + N \mid m \in M\} with induced addition and scalar multiplication, forming an R-module. The natural projection map \pi: M \to M/N given by \pi(m) = m + N is a surjective homomorphism with kernel N. An R-module homomorphism \phi: M \to N is a map preserving addition and scalar multiplication, i.e., \phi(m_1 + m_2) = \phi(m_1) + \phi(m_2) and \phi(r \cdot m) = r \cdot \phi(m) for all r \in R and m_1, m_2 \in M. The kernel \ker \phi = \{m \in M \mid \phi(m) = 0\} is a submodule of M, and the image \operatorname{im} \phi = \{\phi(m) \mid m \in M\} is a submodule of N. The isomorphism theorems for modules mirror those for groups and rings. The first isomorphism theorem states that if \phi: M \to N is a homomorphism, then M / \ker \phi \cong \operatorname{im} \phi. The second asserts that for submodules K \subseteq N \subseteq M, (M/N) / (K/N) \cong M/K. The third follows from the first two, establishing correspondences in quotient structures. These hold for modules over any ring, including commutative ones. A key concept is that of exact sequences of modules. A sequence of R-modules and homomorphisms \cdots \to M_{i-1} \xrightarrow{\phi_{i-1}} M_i \xrightarrow{\phi_i} M_{i+1} \to \cdots is exact at M_i if \operatorname{im} \phi_{i-1} = \ker \phi_i. In particular, a short exact sequence $0 \to K \xrightarrow{i} M \xrightarrow{p} N \to 0 indicates that i is injective with image a submodule isomorphic to K, p is surjective with kernel the image of i, and the sequence captures extensions of modules. For surjective p with kernel K, this yields M / K \cong N via the first isomorphism theorem. Ideals in a commutative ring R are precisely the submodules of the regular module R itself.

Projective Modules

A module P over a commutative ring R is called projective if, whenever there is a surjective R-module homomorphism f: M \to N and a homomorphism g: P \to N, there exists a homomorphism h: P \to M such that f \circ h = g. This lifting property characterizes projective modules via their universal property in the category of R-modules. An equivalent definition is that P is projective if and only if it is a direct summand of some free R-module F, meaning there exists a module K such that F \cong P \oplus K. In particular, every free R-module, such as R^n for n \geq 0, is projective, as it is a direct summand of itself. For example, over a , every finitely generated projective module is free; thus, nonzero ideals, being free of rank 1, are projective if and only if they are free modules. Over an integral domain R with fraction field K, a projective module P has a well-defined rank, which is the dimension of P \otimes_R K as a K-vector space; this rank is constant on the connected components of \operatorname{Spec} R. Projective modules also satisfy \operatorname{Ext}^1_R(P, N) = 0 for every R-module N.

Chain Conditions

Noetherian Rings

A commutative ring R is said to be Noetherian if it satisfies the ascending chain condition on ideals, meaning that every ascending chain of ideals I_1 \subseteq I_2 \subseteq I_3 \subseteq \cdots stabilizes, so there exists some integer k such that I_k = I_{k+1} = I_{k+2} = \cdots. This condition is equivalent to the property that every ideal of R is finitely generated as an R-module. The concept originates from the work of and is fundamental in for ensuring finiteness properties in ideal structures. A key result concerning Noetherian rings is Hilbert's basis theorem, which states that if R is a Noetherian ring, then the polynomial ring R in one indeterminate is also Noetherian. This theorem extends iteratively to show that polynomial rings in any finite number of indeterminates over a Noetherian ring remain Noetherian. For modules over a commutative ring, the notion of Noetherianity generalizes naturally: an R-module M is Noetherian if it satisfies the ascending chain condition on submodules, equivalently, if every submodule of M is finitely generated. Rings themselves are Noetherian precisely when they are Noetherian as modules over themselves. In Noetherian rings, every ideal admits a primary decomposition, meaning that any ideal I can be expressed as a finite intersection I = \mathfrak{q}_1 \cap \cdots \cap \mathfrak{q}_n where each \mathfrak{q}_i is a primary ideal. This decomposition, known as the , allows for the study of associated primes and irredundant decompositions, providing a complete primary factorization analogous to prime factorization in principal ideal domains. The further supports analytic techniques in Noetherian rings: if R is Noetherian, I \subset R is an ideal, and N \subseteq M are finitely generated R-modules, then there exists an integer c \geq 0 such that for all n \geq 0, I^n M \cap N = I^n (I^c M \cap N). This lemma is essential as a precursor to the theory of completions, ensuring that I-adic filtrations behave well on submodules. Prominent examples of Noetherian rings include the ring of integers \mathbb{Z}, which is Noetherian as a principal ideal domain where every ideal is generated by a single element. Polynomial rings k[x_1, \dots, x_n] over a field k are Noetherian by repeated application of . Additionally, power series rings such as k[[x_1, \dots, x_n]] over a field are Noetherian, extending the polynomial case through analogous finiteness arguments.

Artinian Rings

An Artinian ring is a commutative ring that satisfies the descending chain condition on ideals: every descending chain of ideals stabilizes after finitely many steps. Equivalently, there are no infinite strictly descending chains of ideals, and every nonempty set of ideals has a minimal element. This condition ensures that the ring has a "finite depth" in terms of ideal structure, contrasting with Noetherian rings, which satisfy the ascending chain condition; notably, fields satisfy both conditions simultaneously. More generally, an Artinian module over a commutative ring is one that satisfies the descending chain condition on submodules, meaning every descending chain of submodules stabilizes. A ring is Artinian if and only if it is Artinian as a module over itself. Artinian rings are Noetherian and have Krull dimension zero, implying that every prime ideal is maximal. Moreover, they have finite length as modules over themselves: the ring decomposes into a finite composition series of simple modules, with successive quotients being fields. Modules of finite length are precisely those that are both Artinian and Noetherian. A fundamental structure theorem states that every Artinian ring is isomorphic to a finite direct product of local Artinian rings, and this decomposition is unique up to isomorphism and permutation of factors. The number of factors equals the number of maximal ideals, which is finite in Artinian rings. In such rings, the Jacobson radical— the intersection of all maximal ideals—coincides with the nilradical and is nilpotent: there exists a positive integer n such that every product of n elements from the radical is zero. This nilpotency reflects the "smallness" of the ring, as the radical captures the non-semisimple part. Classic examples include the rings \mathbb{Z}/p^n\mathbb{Z} for a prime p and positive integer n, which are local Artinian with maximal ideal (p) and nilpotent radical of index n. Finite direct products of such rings, or more generally of finite field extensions, also yield Artinian rings; for instance, \mathbb{Z}/6\mathbb{Z} \cong \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/3\mathbb{Z} is a product of two fields, hence Artinian. Fields themselves are Artinian, as they have only the zero ideal and the whole ring.

Prime Ideals and Spectrum

Prime and Maximal Ideals

In a commutative ring R with multiplicative identity, a proper ideal \mathfrak{p} \subseteq R is called a prime ideal if, whenever ab \in \mathfrak{p} for a, b \in R, then either a \in \mathfrak{p} or b \in \mathfrak{p}. This condition ensures that the quotient ring R/\mathfrak{p} is an integral domain, meaning it has no zero divisors other than zero. Prime ideals play a central role in commutative algebra, as they capture the "prime" factorization properties analogous to those in the integers. A proper ideal \mathfrak{m} \subseteq R is a maximal ideal if there is no proper ideal of R strictly containing \mathfrak{m}. Equivalently, the quotient R/\mathfrak{m} is a field. Every maximal ideal is prime, since fields are integral domains with no nontrivial ideals. In rings with identity, the existence of maximal ideals follows from Zorn's lemma applied to the partially ordered set of proper ideals ordered by inclusion: every chain of proper ideals has an upper bound (their union, which remains proper), so a maximal element exists, yielding a maximal ideal. This result, known as Krull's theorem, holds for any nonzero commutative ring with identity. The nilradical of R, denoted \sqrt{0}, is the ideal consisting of all nilpotent elements (those x \in R such that x^n = 0 for some positive integer n); it equals the intersection of all prime ideals of R. Similarly, the Jacobson radical of R, denoted J(R), is the intersection of all maximal ideals of R. Since maximal ideals are prime, the nilradical is contained in the Jacobson radical. These radicals provide key information about the "zero divisors" and units in quotients of R. For example, in the ring of integers \mathbb{Z}, the prime ideals are (0) and (p) for each prime number p; the nonzero prime ideals (p) are also maximal, as \mathbb{Z}/(p) \cong \mathbb{Z}/p\mathbb{Z} is a field. In the polynomial ring k[x, y] over a field k, the ideal (x) is prime because k[x, y]/(x) \cong k is an integral domain, but it is not maximal since k is not a field. The set of all prime ideals of R, denoted \operatorname{Spec}(R), forms the foundation for the .

Spectrum and Zariski Topology

The prime spectrum of a commutative ring R, denoted \operatorname{Spec}(R), is the set consisting of all prime ideals of R. The Zariski topology on \operatorname{Spec}(R) is the topology whose closed subsets are the sets of the form V(I) = \{ P \in \operatorname{Spec}(R) \mid P \supseteq I \} for ideals I \subseteq R. These sets satisfy the identities V(I) \cap V(J) = V(IJ), \quad V\left( \bigcup_i I_i \right) = \bigcap_i V(I_i) for any ideals I, J \subseteq R and any family of ideals \{I_i\}. The complements of the sets V(f) for f \in R, known as the basic open sets D(f) = \{ P \in \operatorname{Spec}(R) \mid f \notin P \} = \operatorname{Spec}(R) \setminus V(f), form a basis for the open sets in the Zariski topology. Moreover, D(fg) = D(f) \cap D(g) for any f, g \in R. The space \operatorname{Spec}(R) with the Zariski topology is a spectral space: it is quasi-compact (every open cover has a finite subcover), sober (every irreducible closed subset has a unique generic point), the quasi-compact open subsets form a basis for the topology, and the intersection of any two quasi-compact open subsets is quasi-compact. This structure captures essential geometric properties of the ring R, with points corresponding to prime ideals and the topology reflecting containment relations among them. The ring R can be interpreted as the ring of global regular functions on \operatorname{Spec}(R), yielding a canonical isomorphism R \cong \Gamma(\operatorname{Spec}(R), \mathcal{O}_{\operatorname{Spec}(R)}).

Affine Schemes

Affine Schemes from Rings

In commutative algebra and algebraic geometry, an affine scheme is constructed from a commutative ring R as the pair (\operatorname{Spec} R, \mathcal{O}_{\operatorname{Spec} R}), where \operatorname{Spec} R is the prime spectrum of R and \mathcal{O}_{\operatorname{Spec} R} is its structure sheaf. The prime spectrum \operatorname{Spec} R consists of all prime ideals of R, forming a topological space under the Zariski topology, in which the closed sets are of the form V(I) = \{ \mathfrak{p} \in \operatorname{Spec} R \mid I \subseteq \mathfrak{p} \} for any ideal I \subseteq R. This topology admits a basis of distinguished open sets D(f) = \{ \mathfrak{p} \in \operatorname{Spec} R \mid f \notin \mathfrak{p} \} for f \in R, which cover \operatorname{Spec} R and satisfy D(f) = \operatorname{Spec}(R_f), where R_f denotes the localization of R at the multiplicative set \{1, f, f^2, \dots \}. The structure sheaf \mathcal{O}_{\operatorname{Spec} R} is defined on the basis of distinguished opens by \mathcal{O}_{\operatorname{Spec} R}(D(f)) = R_f, with restriction maps induced by the natural localization homomorphisms R_g \to R_{fg} for compatible opens D(g) \subseteq D(f). This presheaf extends uniquely to a sheaf on \operatorname{Spec} R, making (\operatorname{Spec} R, \mathcal{O}_{\operatorname{Spec} R}) a ringed space whose stalks at points \mathfrak{p} \in \operatorname{Spec} R are the local rings R_\mathfrak{p}, thus yielding a locally ringed space. The global sections satisfy \Gamma(\operatorname{Spec} R, \mathcal{O}_{\operatorname{Spec} R}) = R, ensuring that the ring R can be recovered from the affine scheme. This construction establishes a contravariant functor \operatorname{Spec} from the category of commutative rings to the category of affine schemes, where a ring homomorphism \varphi: R \to S induces a morphism of schemes \operatorname{Spec} \varphi: \operatorname{Spec} S \to \operatorname{Spec} R given by \mathfrak{p} \mapsto \varphi^{-1}(\mathfrak{p}) on points and the sheaf map \mathcal{O}_{\operatorname{Spec} R} \to \varphi_* \mathcal{O}_{\operatorname{Spec} S}. Affine schemes capture geometric objects arising from rings, such as the affine line \mathbb{A}^1_k = \operatorname{Spec} k over a field k, whose points include the generic point (0) and closed points corresponding to maximal ideals (x - a) for a \in k. Every scheme is locally affine, meaning it can be covered by open affine subschemes, with affine schemes serving as the fundamental building blocks.

Morphisms of Affine Schemes

In algebraic geometry, a morphism between affine schemes \operatorname{Spec}(S) \to \operatorname{Spec}(R) is induced by a ring homomorphism \phi: R \to S. Specifically, given such a \phi, the corresponding morphism f: \operatorname{Spec}(S) \to \operatorname{Spec}(R) is defined on points by sending a prime ideal \mathfrak{q} \subset S to its preimage \phi^{-1}(\mathfrak{q}) \subset R, which ensures continuity with respect to the Zariski topology. The morphism f is equipped with a sheaf map f^\#: \mathcal{O}_{\operatorname{Spec}(R)} \to f^{-1} \mathcal{O}_{\operatorname{Spec}(S)}, where on basic open sets D(g) \subset \operatorname{Spec}(R) for g \in R, the map R_g \to S_{\phi(g)} is given by the localization of \phi at g. This construction establishes an anti-equivalence between the category of commutative rings and the opposite category of affine schemes. An affine morphism between schemes is one where the preimage of every affine open subscheme is affine; for morphisms between affine schemes, every such morphism is automatically affine, as \operatorname{Spec}(S) is affine and its preimage under the identity is itself. More generally, affine morphisms arise from quasi-coherent sheaves of algebras on the base, and they are quasi-compact. Flat morphisms of affine schemes correspond to flat ring homomorphisms: a morphism f: \operatorname{Spec}(S) \to \operatorname{Spec}(R) induced by \phi: R \to S is flat if and only if S is a flat R-module via \phi, meaning tensor products preserve exact sequences. Faithfully flat morphisms are flat and surjective, equivalently when \phi makes S faithfully flat over R, ensuring that properties like injectivity of maps can be detected after base change. Open immersions and closed immersions between affine schemes correspond to specific types of ring homomorphisms. A closed immersion \operatorname{Spec}(S) \to \operatorname{Spec}(R) is induced by a surjective homomorphism \phi: R \to S, equivalently S \cong R/I for some ideal I \subset R, with the ideal sheaf \tilde{I} defining the immersion via the sheaf map having kernel \tilde{I}. Conversely, an open immersion \operatorname{Spec}(R_f) \to \operatorname{Spec}(R) for f \in R arises from the injective localization map R \to R_f, embedding the open subscheme D(f) as an affine open. Fiber products of affine schemes \operatorname{Spec}(A) \times_{\operatorname{Spec}(R)} \operatorname{Spec}(B), where A and B are R-algebras via maps R \to A and R \to B, are given by \operatorname{Spec}(A \otimes_R B), preserving the universal property for pullbacks in the category of schemes.

Homomorphisms

Ring Homomorphisms

A ring homomorphism between two commutative rings with unity R and S is a function \phi: R \to S such that \phi(a + b) = \phi(a) + \phi(b), \phi(ab) = \phi(a)\phi(b), and \phi(1_R) = 1_S for all a, b \in R. These conditions ensure that \phi preserves the ring structure, including the additive group operation, the multiplicative operation, and the multiplicative identity. The kernel of \phi, defined as \ker(\phi) = \{ r \in R \mid \phi(r) = 0_S \}, forms an ideal of R. This follows because \ker(\phi) is an additive subgroup closed under multiplication by elements of R, leveraging the commutativity of R. The image \operatorname{im}(\phi) = \{ \phi(r) \mid r \in R \} is a subring of S, as it is closed under addition and multiplication and contains the identity $1_S. A ring homomorphism \phi: R \to S is an isomorphism if it is bijective and its inverse \phi^{-1}: S \to R is also a ring homomorphism. It is an embedding (or monomorphism) if it is injective, which occurs precisely when \ker(\phi) = \{0_R\}. Ring homomorphisms induce module structures: given \phi: R \to S, any S-module M becomes an R-module via the action r \cdot m = \phi(r) m for r \in R, m \in M. This scalar multiplication is well-defined and bilinear, preserving the module axioms. Examples include the inclusion homomorphism \iota: \mathbb{Z} \to \mathbb{Q}, defined by \iota(n) = n/1, which preserves addition, multiplication, and sends $1 to $1. Another is the evaluation homomorphism \operatorname{ev}_a: k \to k for a field k and a \in k, given by \operatorname{ev}_a(f) = f(a), with kernel the principal ideal (x - a). By the first isomorphism theorem for rings, R / \ker(\phi) \cong \operatorname{im}(\phi) as rings.

Finitely Generated Rings

In commutative algebra, a ring S is said to be finitely generated as an R-algebra if there exist finitely many elements x_1, \dots, x_n \in S such that S is generated by these elements over R, meaning every element of S can be expressed as a polynomial in the x_i with coefficients in R. Equivalently, S \cong R[X_1, \dots, X_n]/I for some ideal I \subseteq R[X_1, \dots, X_n]. This presentation arises from the universal property of polynomial rings: given elements x_i \in S, there is a unique R-algebra homomorphism \varepsilon: R\langle X_1, \dots, X_n \rangle \to S sending X_i \mapsto x_i, and S \cong R\langle X_1, \dots, X_n \rangle / \ker(\varepsilon). Such finitely generated R-algebras correspond to morphisms of finite type in scheme theory. Specifically, the induced map \operatorname{Spec}(S) \to \operatorname{Spec}(R) is a morphism of finite type, as S is a quotient of a polynomial ring in finitely many variables over R. A prominent example is the coordinate ring of an affine variety over a field k, given by k[x_1, \dots, x_n]/I(V), where I(V) is the ideal of polynomials in k[x_1, \dots, x_n] vanishing on the variety V \subseteq \mathbb{A}^n_k. These rings are finitely generated over k and encode the algebraic structure of the variety. A key structural result for such algebras is . If S is a finitely generated algebra over a field k, then there exist algebraically independent elements y_1, \dots, y_d \in S (where d is the of S) such that S is integral over the polynomial subring k[y_1, \dots, y_d]. This implies S is a finite module over k[y_1, \dots, y_d], providing a polynomial model for the ring's integral closure. The lemma, originally due to , facilitates dimension theory and reductions in algebraic geometry. Finitely generated algebras over fields also connect to geometric points via . For an algebraically closed field k, the maximal ideals of a finitely generated k-algebra S correspond bijectively to k-rational points in the associated , with the residue field at each maximal ideal being k. This bijection underscores the correspondence between algebraic ideals and geometric loci, though the full theorem addresses and .

Local Rings

Definition and Completions

A local ring is a commutative ring equipped with exactly one maximal ideal. This unique maximal ideal, typically denoted \mathfrak{m}, consists of all non-units in the ring, and the elements outside \mathfrak{m} form the group of units. The quotient R / \mathfrak{m} is a field, known as the residue field of R. Local rings facilitate the local study of global ring properties by focusing on behavior near a specific maximal ideal. One common construction of a local ring is via localization: for a commutative ring A and a maximal ideal \mathfrak{p} \subset A, the localization A_{\mathfrak{p}} is a local ring with unique maximal ideal \mathfrak{p} A_{\mathfrak{p}}. The completion of a local ring (R, \mathfrak{m}) is defined with respect to the \mathfrak{m}-adic topology, where the completion \hat{R} is the inverse limit \hat{R} = \lim_n R / \mathfrak{m}^n. This process equips \hat{R} with a complete metric structure analogous to real analysis, allowing approximation by elements modulo powers of \mathfrak{m}. If R is Noetherian, the natural map R \to \hat{R} is a flat ring homomorphism, and for finitely generated modules, completion preserves exact sequences. The \mathfrak{m}-adic completion satisfies a universal property: the ring homomorphisms \hat{R} \to S to any S are in bijection with the ring homomorphisms \phi: R \to S such that \phi(\mathfrak{m}) is nilpotent in S. A key tool in the study of modules over local rings is . Let R be a local ring with maximal ideal \mathfrak{m} and let M be a finitely generated R-module. If \mathfrak{m} M = M, then M = 0. This lemma implies that if a finitely generated module admits a set of generators that lift to a generating set modulo \mathfrak{m}, then those generators suffice over R. Examples of local rings include discrete valuation rings (DVRs). The ring of p-adic integers \mathbb{Z}_p is the completion of \mathbb{Z} with respect to the ideal (p), forming a DVR with maximal ideal p \mathbb{Z}_p and residue field \mathbb{F}_p. Similarly, the formal power series ring k[] over a field k is a DVR with maximal ideal (x) and residue field k.

Regular Local Rings

A regular local ring is a Noetherian local ring (R, \mathfrak{m}, k) where the Krull dimension \dim R equals the embedding dimension \mathrm{edim}(R), defined as the dimension of the cotangent space \mathfrak{m}/\mathfrak{m}^2 as a vector space over the residue field k = R/\mathfrak{m}. This condition, \dim R = \dim_k (\mathfrak{m}/\mathfrak{m}^2), ensures that the maximal ideal \mathfrak{m} is generated by a regular sequence of length equal to \dim R. Equivalently, \mathrm{edim}(R) = \dim_k (\mathfrak{m}/\mathfrak{m}^2), providing a measure of the minimal number of generators needed for \mathfrak{m}. Classic examples include the localization of a polynomial ring at its maximal ideal, such as k[x_1, \dots, x_d]_{(x_1, \dots, x_d)} over a field k, where the dimension is d and \mathfrak{m} is generated by the d variables forming a regular sequence. Similarly, power series rings like k[[x_1, \dots, x_d]] are regular local rings, with \mathfrak{m} = (x_1, \dots, x_d) generated by a regular sequence of length d. Regular local rings satisfy several strong structural properties. In particular, they are Cohen-Macaulay, meaning the depth of the ring equals its dimension, which follows from the existence of a regular system of parameters generating the maximal ideal. Moreover, regular local rings are integral domains and normal, being integrally closed in their fraction fields, as the associated graded ring with respect to \mathfrak{m} is isomorphic to a polynomial ring over k, which is normal. They are also unmixed, with all minimal prime ideals having height equal to \dim R, reflecting their equidimensionality as Cohen-Macaulay domains.

Homological Algebra

Flat Modules

In commutative algebra, a module M over a commutative ring R is called flat if the tensor product functor -\otimes_R M is exact, meaning that for any short exact sequence of R-modules $0 \to N' \to N \to N'' \to 0, the induced sequence $0 \to N' \otimes_R M \to N \otimes_R M \to N'' \otimes_R M \to 0 is also exact. Equivalently, M is flat if and only if \Tor_1^R(A, M) = 0 for every R-module A. This exactness property implies that flat modules preserve injections: if $0 \to I \to R is an exact sequence of R-modules (where I is an ideal), then $0 \to I \otimes_R M \to R \otimes_R M is exact for any flat M. Free modules provide basic examples of flat modules, as the tensor product with a free module preserves exactness due to the universal property of free constructions. Moreover, every projective module is flat, since projectives are direct summands of free modules and direct summands preserve the exactness of tensor products. A stronger notion is that of a faithfully flat module: an R-module M is faithfully flat if it is flat and, conversely, reflects exactness, meaning that a sequence of R-modules is exact if and only if its tensor product with M is exact. Faithfully flat modules play a key role in and approximations in commutative algebra. Flatness is preserved under base change: if M is a flat R-module and \phi: R \to S is a ring homomorphism, then M \otimes_R S is a flat S-module. Similarly, if \phi: R \to S is flat (i.e., S is flat as an R-module) and N is flat over S, then N is flat over R.

Ext and Tor Functors

In the category of modules over a commutative ring R, the Ext functors provide a measure of the deviation from exactness of the Hom functor. Specifically, for R-modules M and N, the groups \operatorname{Ext}^n_R(M, N) are defined as the right derived functors R^n \operatorname{Hom}_R(M, -)(N), where the zeroth derived functor is \operatorname{Ext}^0_R(M, N) = \operatorname{Hom}_R(M, N). These functors are computed by taking a projective resolution P_\bullet \to M \to 0 of M and applying \operatorname{Hom}_R(P_\bullet, N), yielding \operatorname{Ext}^n_R(M, N) = H^n(\operatorname{Hom}_R(P_\bullet, N)), or equivalently using an injective resolution of N. The Ext groups classify equivalence classes of extensions of modules, with \operatorname{Ext}^1_R(M, N) parametrizing short exact sequences $0 \to N \to E \to M \to 0 up to congruence. Dually, the Tor functors quantify the failure of the tensor product to be right exact. For R-modules M and N, \operatorname{Tor}_n^R(M, N) are the left derived functors L_n(M \otimes_R -)(N), with \operatorname{Tor}_0^R(M, N) = M \otimes_R N. Computation proceeds via a projective resolution P_\bullet \to M \to 0, forming the homology \operatorname{Tor}_n^R(M, N) = H_n(P_\bullet \otimes_R N). These functors are symmetric, \operatorname{Tor}_n^R(M, N) \cong \operatorname{Tor}_n^R(N, M), reflecting the balanced nature of the tensor product over commutative rings. A module M is flat over R if and only if \operatorname{Tor}_1^R(M, N) = 0 for all N. Given a short exact sequence of R-modules $0 \to A \to B \to C \to 0, the derived functors yield long exact sequences. For Ext, this is \cdots \to \operatorname{Ext}^n_R(A, N) \to \operatorname{Ext}^n_R(B, N) \to \operatorname{Ext}^n_R(C, N) \to \operatorname{Ext}^{n+1}_R(A, N) \to \cdots for any N. Similarly, for Tor, \cdots \to \operatorname{Tor}_n^R(N, A) \to \operatorname{Tor}_n^R(N, B) \to \operatorname{Tor}_n^R(N, C) \to \operatorname{Tor}_{n-1}^R(N, A) \to \cdots. These sequences arise from the horseshoe lemma or dimension-shifting arguments and preserve exactness in the module category over commutative rings. A representative example arises from the short exact sequence $0 \to I \to R \to R/I \to 0 for an ideal I \subset R. Applying \operatorname{Hom}_R(-, R) yields a long exact sequence ending with $0 \to \operatorname{Hom}_R(R/I, R) \to \operatorname{Hom}_R(R, R) \to \operatorname{Hom}_R(I, R) \to \operatorname{Ext}^1_R(R/I, R) \to 0, since higher Ext vanish on free modules. This sequence highlights the role of Ext in describing infinitesimal deformations and conormal modules in commutative algebra. Local cohomology with support in an ideal I \subset R can be realized using Ext functors. For an R-module M, the local cohomology groups are H_I^i(M) \cong \varinjlim_n \operatorname{Ext}^i_R(R/I^n, M), where the direct limit is over the directed system induced by the maps R/I^n \to R/I^{n+1}. This identification follows from the derived functor nature of local cohomology as the right derived functors of the I-torsion functor \Gamma_I(M) = \{ m \in M \mid I^k m = 0 \text{ for some } k \}. In certain pullback diagrams of commutative rings, such as R \cong R_1 \times_{R_{12}} R_2, Mayer-Vietoris long exact sequences exist for both Tor and Ext. For instance, if A, B \subset M are submodules with M = A + B, there is an exact sequence \cdots \to \operatorname{Tor}_n^R(M, N) \to \operatorname{Tor}_n^R(A, N) \oplus \operatorname{Tor}_n^R(B, N) \to \operatorname{Tor}_n^R(A \cap B, N) \to \operatorname{Tor}_{n-1}^R(M, N) \to \cdots, analogous to the topological Mayer-Vietoris sequence. A similar sequence holds for Ext by contravariant application.

Dimension and Nullstellensatz

Krull Dimension

The Krull dimension of a commutative ring R, denoted \dim R, is the supremum of the lengths of all strictly ascending chains of prime ideals in R. Specifically, if there exists a chain \mathfrak{p}_0 \subsetneq \mathfrak{p}_1 \subsetneq \cdots \subsetneq \mathfrak{p}_d of distinct prime ideals, the length of the chain is d, and \dim R is the least upper bound of such lengths over all possible chains (which may be infinite). This definition quantifies the "size" of the prime spectrum \operatorname{Spec}(R) as a partially ordered set under inclusion. For an R-module M, the Krull dimension \dim M is defined as the supremum of \dim R / \mathfrak{p} where \mathfrak{p} ranges over the associated primes of M. This extends the ring case, as \dim R = \dim R / (0) when (0) is prime, and captures the dimensional complexity of the module via its "minimal" prime supports. For finitely generated modules over Noetherian rings, this coincides with the dimension of the support \operatorname{Supp}(M). Several key properties characterize the Krull dimension. For any commutative ring R, adjoining an indeterminate yields \dim R = \dim R + 1, reflecting the increase in "transcendence degree" or geometric dimension. In a local ring (R, \mathfrak{m}), the dimension equals the height of the maximal ideal \mathfrak{m}, i.e., \dim R = \ht \mathfrak{m}, which is the length of the longest chain of primes contained in \mathfrak{m}. Integral extensions preserve dimension via the going-up and going-down theorems. If R \subseteq S is an integral extension of commutative rings, then for any chain of primes \mathfrak{p}_0 \subsetneq \cdots \subsetneq \mathfrak{p}_l in R, there exists a corresponding chain \mathfrak{q}_0 \subsetneq \cdots \subsetneq \mathfrak{q}_l in S with \mathfrak{q}_i \cap R = \mathfrak{p}_i for each i (going-up theorem); thus, \dim S = \dim R. Under additional assumptions (e.g., R normal and both domains), the going-down theorem ensures chains can be extended "downward," equating heights: \ht \mathfrak{q} = \ht (\mathfrak{q} \cap R). A fundamental inequality relates quotients to heights: for any ideal I in R, \dim R / I \geq \dim R - \ht I, where \ht I is the maximum height of primes contained in I (Krull's principal ideal theorem in the general case). Equality holds under Noetherian assumptions for principal ideals. Examples illustrate these concepts. The ring of integers \mathbb{Z} has \dim \mathbb{Z} = 1, as the longest prime chain is (0) \subsetneq (p) for prime p. For a field k, the polynomial ring k[x, y] has \dim k[x, y] = 2, via chains like (0) \subsetneq (x) \subsetneq (x, y).

Hilbert's Nullstellensatz

Hilbert's Nullstellensatz establishes a profound correspondence between ideals in polynomial rings over algebraically closed fields and geometric varieties in affine space, bridging commutative algebra and algebraic geometry. The theorem, first proved by David Hilbert in 1893, asserts that algebraic conditions on polynomials translate directly into geometric properties of their zero sets. It exists in weak and strong forms, with the weak version characterizing maximal ideals and the strong version relating radicals of ideals to vanishing sets. The weak Nullstellensatz states that if k is an algebraically closed field and R = k[x_1, \dots, x_n] is the polynomial ring in n variables, then every maximal ideal \mathfrak{m} of R has the form \mathfrak{m} = (x_1 - a_1, \dots, x_n - a_n) for some point a = (a_1, \dots, a_n) \in k^n. This implies that every proper ideal of R has at least one zero in k^n, or equivalently, a system of polynomials with no common zero generates the unit ideal. Geometrically, maximal ideals correspond bijectively to points in affine n-space over k, establishing that points are the "irreducible" geometric objects dual to maximal ideals. The strong Nullstellensatz extends this by stating that for a proper ideal I \subseteq R = k[x_1, \dots, x_n] and any f \in R that vanishes on the variety V(I) = \{ p \in k^n \mid g(p) = 0 \ \forall g \in I \}, there exists a positive integer m such that f^m \in I. Equivalently, the \sqrt{I} = \{ f \in R \mid f^m \in I \ \text{for some} \ m > 0 \} is precisely the ideal I(V(I)) consisting of all polynomials vanishing on V(I). This yields I(V(J)) = \sqrt{J} for any ideal J, ensuring that geometric varieties are determined by radical ideals and vice versa. Key consequences include the realization that varieties over algebraically closed fields consist of finitely many points only when the defining ideal has codimension equal to the ambient dimension, as in zero-dimensional cases where V(I) is finite. More broadly, the theorem links the algebraic dimension of finitely generated rings over k—measured by transcendence degree of the fraction field—to the geometric dimension of varieties, providing a foundation for studying transcendence in algebraic extensions. A standard proof of the weak Nullstellensatz relies on Zariski's , which asserts that if k is algebraically closed and L is a finitely generated as a k-, then L = k. To see this, suppose \mathfrak{m} is maximal in R; then R/\mathfrak{m} is a finitely generated over k, so by Zariski's , R/\mathfrak{m} \cong k, implying \mathfrak{m} = (x_1 - a_1, \dots, x_n - a_n) via the evaluation map at some a \in k^n. Zariski's itself follows from choosing a transcendence basis and using the closure properties of finite extensions over algebraically closed fields. The strong Nullstellensatz can be deduced from the weak version using Noether normalization, which guarantees that a finitely generated k- integral over a subring admits a finite structure. Alternatively, Rabinowitsch's trick reduces it to the weak case: if f vanishes on V(I), adjoin a new y and consider the (I, 1 - y f) in R; by the weak , this ideal is the whole ring, yielding polynomials whose combination shows some power of f lies in I. This approach highlights the theorem's reliance on finite generation over k, as discussed in the context of finitely generated rings.

Generalizations

Graded Rings

A graded commutative ring is a commutative ring R equipped with a direct sum decomposition R = \bigoplus_{n \in \mathbb{Z}} R_n as abelian groups, such that the multiplication satisfies R_m \cdot R_n \subseteq R_{m+n} for all m, n \in \mathbb{Z}, and typically R_n = 0 for n < 0, with the unit element in R_0. Elements of R_n are called homogeneous of degree n, and for homogeneous elements f \in R_m and g \in R_n, the degree of their product satisfies \deg(fg) = \deg f + \deg g = m + n. A homogeneous ideal in a graded ring R is an ideal generated by homogeneous elements, and the set of such ideals forms a distinguished subset of the ideals of R. The Proj construction associates to a graded commutative ring R (with R_+ = \bigoplus_{n > 0} R_n finitely generated as an R_0-) the scheme Proj(R), which is the quotient of Spec(R) excluding the irrelevant ideal V(R_+), equipped with the restricted to homogeneous prime ideals not containing R_+. This yields a projective scheme over Spec(R_0), providing a geometric object that complements affine s by incorporating . For a finitely generated graded M over a graded Noetherian commutative ring R, the Hilbert function h_M(n) counts the of the degree-n component: h_M(n) = \dim_k (M_n) when R_0 = k is a , measuring the growth of M. For sufficiently large n, this function coincides with a p_M(t) \in \mathbb{Q}, called the Hilbert , whose equals the of the support of M minus one, providing invariants like the multiplicity and of associated projective varieties. A canonical example is the polynomial ring k[x_0, \dots, x_d] over a k, graded by total degree, where homogeneous ideals correspond to projective subvarieties, and Proj(k[x_0, \dots, x_d]) recovers the \mathbb{P}^d_k. Graded rings thus underpin the study of projective varieties, enabling the embedding of algebraic varieties into projective space via and facilitating tools like the for parametrizing subschemes of fixed Hilbert polynomial.

Derived Commutative Rings

In the context of homological algebra, derived commutative rings generalize classical commutative rings by incorporating differential structures and homotopies, allowing for a treatment of non-flat modules and derived functors within the ring-theoretic framework. A differential graded (dg) commutative ring over a base commutative ring k is a \mathbb{Z}-graded k-algebra A^\bullet = \bigoplus_{n \in \mathbb{Z}} A^n, equipped with a differential d: A^\bullet \to A^{\bullet + 1} satisfying d^2 = 0 and the Leibniz rule d(ab) = d(a)b + (-1)^{\deg(a)} a d(b), where the multiplication is graded-commutative, meaning ab = (-1)^{\deg(a)\deg(b)} ba for homogeneous elements a, b. This structure ensures that the multiplication is compatible with the grading and differential, forming a chain complex that models infinitesimal extensions and deformations of rings. A key operation in this setting is the derived tensor product \otimes^L, which resolves the issue of non-exactness in the ordinary over non-flat modules. For two dg-modules M, N over a dg commutative ring A, the derived tensor product M \otimes^L_A N is computed by taking flat (or projective) resolutions and applying the underived tensor product, yielding a bifunctor on the D(A) that preserves homotopical information. The derived spectrum, denoted \operatorname{Spec}^L(A), assigns to a dg commutative ring A an affine derived scheme, representing the functor that sends a simplicial commutative ring B to the derived mapping space \operatorname{Map}(A, B) in the \infty-category of simplicial rings, thus bridging classical geometry with . In , derived commutative rings connect to E_\infty ring spectra, which are commutative monoids in the \infty-category of spectra, providing a spectral analogue where classical s embed via Eilenberg-MacLane spectra; this framework underlies by allowing "derived stacks" to model intersections and quotients that fail to be transverse in the classical setting. For instance, chain complexes of k-vector spaces form dg-modules over the dg ring k (concentrated in degree 0), and their endomorphism dg algebras can be commutative up to , illustrating how derived structures capture resolutions like the . Applications of derived commutative rings appear prominently in derived algebraic geometry, where they enable the study of derived stacks—higher stacks whose "points" are given by dg or simplicial commutative rings—facilitating computations of virtual intersections and moduli spaces that are invisible classically, such as in the deformation theory of schemes. A central tool here is Hochschild cohomology, which governs derived deformations: for a commutative ring R, the Hochschild cohomology groups satisfy HH^*(R) = \operatorname{Ext}^*_{R^e}(R, R), where R^e = R \otimes_k R^{op} is the enveloping algebra, measuring infinitesimal extensions up to homotopy and linking to the tangent complex in derived geometry.

Applications

Algebraic Geometry

In algebraic geometry, commutative rings serve as the foundational algebraic structures for describing geometric objects such as varieties and schemes. The coordinate ring of an affine variety V \subseteq k^n, where k is an , is defined as k[V] = k[x_1, \dots, x_n]/I(V), where I(V) is the ideal of polynomials vanishing on V. This ring encodes the regular functions on V and captures its geometric properties through ideals and quotients. Affine schemes, constructed as the of such rings, provide a more general framework that unifies varieties with non-reduced structures. Hilbert's basis theorem ensures that every ideal in the polynomial ring k[x_1, \dots, x_n] is finitely generated, implying that any affine variety can be defined by a finite set of polynomial equations. This finiteness is crucial for computational aspects of and for proving that varieties are zero loci of finite systems. Bézout's theorem, which states that two plane curves of degrees d and e intersect in de points (counted with multiplicity) over an algebraically closed field, can be proved using resultants in the polynomial ring k[x, y]. The resultant of two polynomials provides a condition for common roots, allowing the intersection multiplicity to be computed algebraically. A landmark result in algebraic geometry is Hironaka's theorem on the resolution of singularities for varieties over fields of characteristic zero, which asserts that any such variety admits a resolution—a proper birational morphism from a smooth variety—achieved through a finite sequence of blow-ups. This 1964 result relies on commutative algebra techniques, such as normalization and valuation theory in rings, to systematically eliminate singularities. In modern , étale cohomology provides a tool for studying the topology of varieties over arbitrary fields, using the étale topology on schemes associated to commutative rings. This cohomology theory, which generalizes classical sheaf cohomology, computes invariants like the étale fundamental group and is essential for the . For a projective C embedded in , the Hilbert polynomial P_C(t) of its coordinate (or more precisely, the twisting sheaf) is linear: P_C(t) = d t + 1 - g, where d is the and g is the arithmetic . Thus, the genus is recovered as g = 1 - P_C(0), linking algebraic invariants of the ring to geometric properties of the .

Number Theory

In number theory, commutative rings play a central role through the study of , which provide a framework for understanding unique factorization in rings of integers of number fields. A is defined as an integrally closed Noetherian of 1, where every nonzero is maximal. Equivalently, it is a Noetherian domain in which every nonzero fractional ideal is invertible, ensuring that every nonzero ideal factors uniquely as a product of . This structure generalizes domains while restoring unique factorization at the level of ideals rather than elements. are a special case of , distinguished by their arithmetic properties that facilitate the analysis of prime factorization in algebraic integers. A key example arises in , where the \mathcal{O}_K of a number field K (the of \mathbb{Q} in a finite extension) forms a . For instance, in quadratic fields K = \mathbb{Q}(\sqrt{d}) with d, \mathcal{O}_K = \mathbb{Z}[\sqrt{d}] if d \equiv 2,3 \pmod{4}, or \mathbb{Z}\left[\frac{1+\sqrt{d}}{2}\right] otherwise, and these rings satisfy the defining properties of Dedekind domains. In such rings, every nonzero ideal factors uniquely into prime ideals, allowing the extension of unique from \mathbb{Z} to more general arithmetic settings. To quantify ideals, the norm N(I) of a nonzero ideal I in a Dedekind domain R with fraction field K is defined as the cardinality of the finite quotient R/I, or equivalently for principal ideals (a), N((a)) = |N_{K/\mathbb{Q}}(a)| where N_{K/\mathbb{Q}} is the field norm. This norm is multiplicative, N(IJ) = N(I)N(J), and for prime ideals \mathfrak{p} lying over a rational prime p, N(\mathfrak{p}) = p^f with f the inertial degree. The ideal class group, denoted \mathrm{Pic}(R) or \mathrm{Cl}(R), measures the failure of unique element factorization and is the quotient of the group of fractional ideals by the subgroup of principal fractional ideals. For \mathcal{O}_K, \mathrm{Cl}(\mathcal{O}_K) is finite, and its order, the class number h_K, governs the complexity of solving Diophantine equations in K. In field extensions L/K of number fields, with integral closures \mathcal{O}_L and \mathcal{O}_K both Dedekind domains, ramification describes how primes of \mathcal{O}_K decompose in \mathcal{O}_L. A prime \mathfrak{p} of \mathcal{O}_K ramifies if it divides the ideal \mathfrak{d}_{L/K}, which is the norm of the different ideal \mathfrak{D}_{L/K} measuring the inseparability and wild ramification. The \Delta_{L/K} is the ideal norm N_{L/K}(\mathfrak{D}_{L/K}) in \mathbb{Z}, and primes dividing \Delta_{L/K} are exactly those that ramify. This framework quantifies how extensions distort the arithmetic of primes, essential for computing class numbers and units. Historically, commutative ring theory advanced through efforts to prove , x^n + y^n = z^n has no positive integer solutions for n > 2. In the , introduced ideal numbers in the of cyclotomic fields \mathbb{Q}(\zeta_p) for odd primes p, proving the theorem for "regular" primes where the class number is not divisible by p, using unique ideal factorization in these Dedekind domains. This approach highlighted the role of class groups in arithmetic progress and influenced the development of .