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Prime ideal

In ring theory, a prime ideal of a commutative ring R with identity is a proper ideal P \neq R such that whenever the product ab \in P for elements a, b \in R, then at least one of a \in P or b \in P.^[1]^[2] This property is equivalent to the quotient ring R/P forming an integral domain, meaning it has no zero divisors.^[1]^[3] Prime ideals generalize the concept of prime numbers from the integers to ideals in arbitrary rings, where the principal ideals generated by prime integers in \mathbb{Z}, such as (p) for a prime p, serve as prototypical examples.^[4] In polynomial rings like \mathbb{Z}, examples include the zero ideal (0), principal ideals (p) for prime p, (x), and (p, x), with the latter being maximal.^[3] The zero ideal is prime in any integral domain, while in fields, the only proper ideal is the zero ideal, which is prime.^[2] Every maximal ideal is prime, but the converse does not hold; for instance, (x) in \mathbb{R} is prime yet not maximal, as \mathbb{R}/(x) \cong \mathbb{R} is an integral domain but not a field.^[2]^[3] Prime ideals play a central role in commutative algebra, forming the basis for the spectrum of a ring, which endows the set of prime ideals with a Zariski topology used to model geometric objects like affine varieties.^[3] Their study underpins key results such as the existence of maximal ideals via Zorn's lemma, ensuring every nonzero ring contains at least one prime ideal.^[5] In algebraic geometry, prime ideals correspond to irreducible varieties, highlighting their foundational importance in bridging algebra and geometry.^[1]

Prime ideals in commutative rings

Definition

In commutative ring theory, a prime ideal of a commutative ring R with identity is a proper ideal P \neq R such that whenever the product ab \in P for elements a, b \in R, then at least one of a \in P or b \in P.^[1] This property ensures that the ideal captures a form of "indivisibility" analogous to prime numbers. The definition requires commutativity to make the elementwise condition meaningful, as noncommutative rings use a different formulation involving ideal products.

Equivalent characterizations

A prime ideal P in a commutative ring R with unity admits several equivalent characterizations that provide alternative perspectives on its defining property that whenever ab \in P, then a \in P or b \in P. One standard characterization is that P is prime if and only if the quotient ring R/P is an integral domain.^[5] To see this equivalence, assume first that P is prime. Suppose \overline{a} \overline{b} = 0 in R/P, so ab \in P. By primeness of P, either a \in P or b \in P, hence \overline{a} = 0 or \overline{b} = 0. Thus, R/P has no zero divisors. Moreover, the unity \overline{1} is preserved in the quotient (since $1 \notin P, as P is proper), making R/P an integral domain. Conversely, assume R/P is an integral domain. If ab \in P, then \overline{a} \overline{b} = 0 in R/P, so \overline{a} = 0 or \overline{b} = 0, meaning a \in P or b \in P. Commutativity ensures the ring operations in R/P align with those of an integral domain, while the unity condition guarantees the quotient is unital.^[5] Another equivalent characterization involves localization: P is prime if and only if the localization R_{(P)} = S^{-1}R, where S = R \setminus P is the multiplicative set of elements outside P, is an integral domain.^[6] For the forward direction, assume P is prime, so R/P is an integral domain by the previous characterization. The natural map R \to R_{(P)} has kernel K = \{ r \in R \mid \exists s \in S, \, sr = 0 \}. For any r \in K, sr = 0 with s \notin P implies \overline{s} \overline{r} = 0 in R/P, and since \overline{s} \neq 0 and R/P has no zero divisors, \overline{r} = 0, so r \in P. Thus, K \subseteq P. In fact, K = P under these conditions, and R_{(P)} \cong (R/P)_{(0)}, the localization of the domain R/P at its nonzero elements, which is the field of fractions of R/P and hence an integral domain. For the converse, assume R_{(P)} is an integral domain. The maximal ideal of R_{(P)} is m = P R_{(P)} = \{ p/s \mid p \in P, s \in S \}, and the contraction of m back to R is \{ r \in R \mid r/1 \in m \} = \{ r \in R \mid \exists s \in S, \, sr \in P \}. Since R_{(P)} / m is a field (hence an integral domain), m is prime in R_{(P)}. If ab \in P, then (a/1)(b/1) = ab/1 \in m, so a/1 \in m or b/1 \in m because m is prime, implying a \in P or b \in P. The unital and commutative structure ensures the localization preserves these properties without introducing extraneous zero divisors.^[6] A more general characterization uses arbitrary multiplicative sets: P is prime if and only if for every multiplicative set S \subseteq R with S \cap P = \emptyset, the localization S^{-1}R has no zero divisors (i.e., is an integral domain).^[7] If P is prime, then R/P is an integral domain, and the image S' of S in R/P is a multiplicative set consisting of nonzero elements (since S \cap P = \emptyset). The localization S^{-1}R maps to (S')^{-1}(R/P), which is a subring of the field of fractions of the domain R/P and thus an integral domain; by faithfulness of the localization functor under these conditions, S^{-1}R is also an integral domain. Conversely, taking S = R \setminus P yields that R_{(P)} is an integral domain, reducing to the previous characterization. The unity in R ensures $1 \in S for such sets, and commutativity allows the fraction field construction to apply directly.^[7]

Examples

In the ring of integers \mathbb{Z}, the prime ideals are the principal ideals (p) generated by a prime number p, along with the zero ideal (0). For instance, (2) is prime because if ab \in (2), then 2 divides ab, so 2 divides a or b, meaning a \in (2) or b \in (2). The quotient \mathbb{Z}/(p) \cong \mathbb{Z}/p\mathbb{Z} is a field, hence an integral domain.^[3] In the polynomial ring k over a field k, the prime ideals are the zero ideal (0) and the principal ideals (f) where f is an irreducible polynomial. For example, (x) is prime, as k/(x) \cong k is a field (thus an integral domain). The zero ideal is prime since k is an integral domain.^[3] In \mathbb{Z}, examples include (0), (p) for prime p, (x), and (p, x). The ideal (p, x) is maximal (hence prime), with \mathbb{Z}/(p, x) \cong \mathbb{F}_p, a field. The ideal (x) is prime but not maximal, as \mathbb{Z}/(x) \cong \mathbb{Z}, an integral domain but not a field.^[3] In any integral domain, the zero ideal (0) is prime, since the ring itself has no zero divisors. In a field, the zero ideal is the only proper ideal and is prime.^[2]

Non-examples

In the ring of integers \mathbb{Z}, the principal ideal (4) generated by 4 is not prime. To see this, note that $2 \cdot 2 = 4 \in (4), but $2 \notin (4), violating the condition that if the product of two elements lies in the ideal, then at least one must lie in it.^[5] Equivalently, the quotient ring \mathbb{Z}/(4) has zero divisors, such as the images of 2 and 2 whose product is zero, so it is not an integral domain.^[5] Similarly, in \mathbb{Z}, the ideal (6) is not prime because $2 \cdot 3 = 6 \in (6), yet neither 2 nor 3 belongs to (6).^[5] The quotient \mathbb{Z}/(6) is isomorphic to \mathbb{Z}/6\mathbb{Z}, which contains zero divisors like the classes of 2 and 3, confirming it fails to be an integral domain.^[5] In the polynomial ring k[x, y] over a field k, the ideal (x^2, y) is not prime. Here, x \cdot x = x^2 \in (x^2, y), but x \notin (x^2, y) since x cannot be expressed as a combination of x^2 and y with coefficients in k[x, y].^[5] The quotient k[x, y]/(x^2, y) has zero divisors, as the image of x squared is zero while the image of x is nonzero, so it is not an integral domain.^[5] Consider the ring \mathbb{Z}/6\mathbb{Z}, which has zero divisors. Its zero ideal (0) is not prime because \overline{2} \cdot \overline{3} = \overline{0} \in (0), but neither \overline{2} nor \overline{3} is zero in the ring.^[5] This ring is not an integral domain, directly showing the zero ideal fails the primeness condition.^[5]

Algebraic properties

In commutative rings, the nilradical of a ring R, denoted \mathfrak{N}(R), which consists of all nilpotent elements, equals the intersection of all prime ideals of R^[8]. More generally, for any ideal I of R, the radical \sqrt{I} is the intersection of all prime ideals containing I^[8]. These equalities highlight the central role of prime ideals in determining radical structure. The intersection of any collection of prime ideals in a commutative ring is itself a radical ideal, since its radical coincides with the intersection itself, but such an intersection is not necessarily prime unless the collection consists of a single prime ideal. For instance, the intersection of distinct maximal ideals yields a proper ideal that fails the primeness condition, as the quotient ring is a nontrivial product of fields. Regarding extensions, if P is a prime ideal in a commutative ring R and S is a ring containing R as a subring, the extended ideal PS is not necessarily prime in S; for example, it may decompose into a product of distinct primes in certain polynomial extensions^[9]. However, in the special case where S is integral over R, the lying-over theorem ensures the existence of at least one prime ideal Q in S such that Q \cap R = P, though PS itself is typically the intersection of all such primes lying over P and thus radical but not necessarily prime^[10]. A key tool for navigating containment relations among ideals is the prime avoidance lemma, which states that if J is an ideal in a commutative ring R and I_1, \dots, I_r are ideals such that J \not\subseteq I_i for each i, with all but at most two of the I_i being prime, then there exists an element x \in J avoiding all the I_i, i.e., x \not\in I_i for all i = 1, \dots, r^[11]. This lemma is particularly useful for proving that certain ideals generated by fewer elements than the height of a prime cannot be contained in that prime. Finally, chains of prime ideals provide a measure of the complexity of the ring's ideal structure via the Krull dimension, defined as the supremum of the lengths of all strictly increasing chains of prime ideals in R^[12]. For a prime ideal P, its height is the length of the longest chain of primes contained in P, and the dimension of R equals the supremum of the heights of its maximal ideals^[12].

Geometric interpretations

In algebraic geometry, the geometric interpretation of prime ideals arises through the affine scheme \operatorname{Spec}(R), where R is a commutative ring. The points of \operatorname{Spec}(R) are the prime ideals of R, and this space captures the geometric structure dual to the ring R. Specifically, each prime ideal \mathfrak{p} \in \operatorname{Spec}(R) corresponds to an irreducible closed subscheme, which generalizes the notion of an irreducible affine variety when R is the coordinate ring of such a variety. This correspondence allows prime ideals to represent the "generic points" of irreducible components in the geometric object associated to R.^[13] Hilbert's Nullstellensatz provides the foundational link between ideals and varieties over an algebraically closed field k. It establishes that for a proper ideal I \subset k[x_1, \dots, x_n], the vanishing set V(I) is nonempty, and the radical \sqrt{I} = I(V(I)). Moreover, V(I) decomposes into irreducible components corresponding to the minimal prime ideals containing I, with each such prime ideal \mathfrak{p} defining an irreducible variety V(\mathfrak{p}). Thus, prime ideals delineate the irreducible building blocks of algebraic varieties, ensuring that radical ideals capture zero loci precisely.^[14] The height of a prime ideal \mathfrak{p} in R, defined as the supremum of lengths of chains of prime ideals descending from \mathfrak{p}, corresponds geometrically to the codimension of the associated irreducible variety V(\mathfrak{p}) in the ambient space. For instance, in the polynomial ring k[x, y] over an algebraically closed field k, the prime ideal (x) has height 1 and corresponds to the y-axis, an irreducible curve (codimension 1) in the affine plane \mathbb{A}^2_k. This alignment underscores how the arithmetic of ideals mirrors the dimensionality of geometric objects. In the Zariski topology on \operatorname{Spec}(R), the closed sets are of the form V(I) = \{\mathfrak{p} \in \operatorname{Spec}(R) \mid I \subseteq \mathfrak{p}\} for ideals I \subseteq R. Each irreducible closed subset has a unique generic point, which is the minimal prime ideal in that subset, representing the entire irreducible component. This structure endows \operatorname{Spec}(R) with a sober topology, where prime ideals serve as the "points at infinity" or generic loci of varieties, facilitating the study of schemes beyond classical varieties.^[15]

Prime ideals in noncommutative rings

Definition

In the context of noncommutative ring theory, the definition of a prime ideal extends the commutative notion but replaces the condition on products of elements with one involving products of ideals, due to the failure of commutativity to preserve analogous properties for elements. A two-sided ideal P in a noncommutative ring R with identity is defined to be prime if P \neq R and, for any two-sided ideals A and B of R, the inclusion AB \subseteq P implies either A \subseteq P or B \subseteq P, where the ideal product AB is the set of all finite sums of elements of the form ab with a \in A and b \in B.^[16] This formulation emphasizes the role of ideal products over elementwise multiplication, as the latter does not yield a satisfactory generalization in the noncommutative setting. In contrast to the commutative case—where a prime ideal satisfies the elementwise condition that ab \in P implies a \in P or b \in P—the noncommutative definition avoids this because, without commutativity, ab \in P does not force a \in P or b \in P for arbitrary elements, even when P is prime.^[16] The requirement that ideals be two-sided ensures the product AB is well-defined and symmetric in structure, aligning with the two-sided nature of ideals in noncommutative rings. This adaptation was introduced in the late 1940s to facilitate the study of ideal structure beyond commutative rings.^[17]^[16]

Examples

In the full matrix ring M_n(R) over a commutative ring R with n \geq 2, the two-sided prime ideals are precisely those of the form M_n(\mathfrak{p}), where \mathfrak{p} is a prime ideal of R. For instance, if R = k is a field, then M_n(k) is a simple Artinian ring, so the zero ideal is the unique proper two-sided ideal and is prime, as the absence of nonzero proper two-sided ideals ensures that if AB \subseteq (0) for two-sided ideals A, B, then A = (0) or B = (0). Minimal left or right ideals in M_n(k), such as the ideals generated by a single matrix with a single nonzero entry, are not two-sided and thus cannot be prime in the two-sided sense; prime ideals instead arise from the simple Artinian structure itself.^[4] In the Weyl algebra A_1(k) = k\langle x, \partial \rangle over a field k of characteristic zero, defined by the relation \partial x - x \partial = [1](/page/1), the ring is simple, so the zero ideal is prime. More generally, prime ideals correspond to the annihilators of irreducible representations, which in this case coincide with the zero ideal since all proper ideals are zero; this reflects the ring's simplicity, where any nonzero two-sided ideal generates the whole algebra. In group rings [kG](/page/KG) over a field k, where G is a group, the primitive ideals—annihilators of irreducible kG-modules—are always prime ideals. For example, if G is finite and k algebraically closed, these primitive ideals are the kernels of irreducible representations and satisfy the prime condition because the quotient kG / P is a primitive ring, hence prime (i.e., if AB \subseteq P for ideals A, B in kG, then A \subseteq P or B \subseteq P). Verification follows from the fact that primitive rings have no nonzero ideals whose product is zero. In the free algebra k\langle x, y \rangle over a field k, the two-sided ideal (x) generated by x is prime. The quotient k\langle x, y \rangle / (x) \cong k\langle y \rangle is a domain (as free algebras on one generator have no zero divisors), so (x) satisfies the prime property: if AB \subseteq (x) for two-sided ideals A, B, then the images of A and B in the quotient multiply to zero, implying one is zero in the quotient, hence A \subseteq (x) or B \subseteq (x). This can be verified directly using the free basis of monomials, where elements not in (x) involve powers of y without x, and their products avoid (x) unless one factor is contained therein.

Distinguishing features

In noncommutative rings, a prime ideal P distinguishes itself through the property that the quotient ring R/P is a prime ring, meaning it contains no zero-divisor ideals: whenever A and B are nonzero two-sided ideals of R/P, their product AB is nonzero.^[18] This ideal-theoretic condition contrasts with the element-wise zero-divisor absence in commutative quotients, emphasizing the role of bilateral ideal products in noncommutative settings. Prime ideals in noncommutative rings connect deeply to module theory, particularly via faithful modules. Specifically, if a prime ring R/P admits a nonzero socle \mathrm{Soc}(R/P), then this socle forms a simple faithful left module over R/P, rendering the quotient left primitive.^[19] In such prime rings, the annihilator of any nonzero left ideal is zero, ensuring that the ring acts without kernel on its ideals and underscoring the absence of nontrivial annihilator ideals.^[18] A key structural result is Goldie's theorem, which implies that artinian prime rings possess finite uniform dimension: the maximal length of a direct sum of nonzero right ideals is finite and invariant.^[20] In fact, left artinian prime rings are precisely the simple artinian rings, such as matrix rings over division rings, where the uniform dimension equals the matrix degree.^[21] Noncommutative localization at a prime ideal P—constructing a ring inverting elements outside P—exists only under the Ore condition: for any r \in R \setminus P and s \in P, there exist r' \in R \setminus P and s' \in P such that r s' = s r'.^[22] This requirement makes the process more intricate than in commutative rings, where localization always succeeds without additional hypotheses.^[23] Unlike commutative prime ideals, which behave "elementarily" by yielding integral domains, noncommutative primes demand scrutiny of the entire ideal lattice in the quotient, as primeness hinges on ideal containment rather than scalar multiplication.^[18] For instance, as illustrated in examples like full matrix rings over division rings, these quotients are often simple, highlighting the structural rigidity imposed by noncommutativity.

Relations to other ideals and structures

Connection to maximal ideals

In any ring with unity, every maximal ideal is a prime ideal. To see this, consider a maximal two-sided ideal M in a ring R. The quotient R/M is a simple ring, meaning it has no nontrivial two-sided ideals. If AB \subseteq M for two-sided ideals A, B of R not contained in M, then the images of A and B in R/M would generate a nontrivial ideal, contradicting simplicity unless one of them is zero, which implies A \subseteq M or B \subseteq M. Thus, M satisfies the definition of a prime ideal.^[21] In the commutative case, the implication follows more directly from the quotient structure. For a commutative ring R with unity and a maximal ideal M, the quotient R/M is a field, hence an integral domain. An ideal P of R is prime if and only if R/P is an integral domain, so M is prime.^[3] The converse does not hold in general: there exist prime ideals that are not maximal. In commutative rings, this is evident from examples like the zero ideal in \mathbb{Z}, which is prime but properly contained in maximal ideals such as (2). In noncommutative rings, the situation is similar; for instance, the zero ideal is prime in any prime ring (a ring where the zero ideal is prime), but such rings need not be simple, so the zero ideal is not maximal. Examples include certain Ore extensions or free algebras over fields that are prime but possess nontrivial ideals. Maximality implies primeness holds universally in rings with unity, but the reverse fails due to the possibility of infinite ascending chains of ideals or non-simple structures.^[24]^[21] In commutative rings, the prime ideals form the points of the spectrum \operatorname{Spec}(R), equipped with the Zariski topology, while the maximal ideals form the closed points, comprising the maximal spectrum \operatorname{MaxSpec}(R) as a subspace. The set of all prime ideals is dense in this topology on \operatorname{MaxSpec}(R) in the sense that every maximal ideal is the closure of the generic points (minimal primes) it contains, reflecting the hierarchical structure where non-maximal primes "specialize" to maximals.^[3] Every prime ideal is contained in some maximal ideal, assuming the axiom of choice. This follows from Zorn's lemma applied to the partially ordered set of proper ideals containing a given prime ideal P, ordered by inclusion. Every chain in this set has an upper bound (the union), so a maximal element exists, which must be a maximal ideal of R properly containing P. Without choice, the existence may fail, but in standard settings with unity, it ensures that prime ideals extend to maximals.^[3]

Relation to primary ideals

In commutative algebra, a proper ideal Q in a commutative ring R is called primary if whenever ab \in Q for elements a, b \in R, then either a \in Q or b^n \in Q for some positive integer n.^[25] This condition ensures that the quotient ring R/Q has the property that every zero-divisor is nilpotent.^[26] Every prime ideal is primary, as the definition simplifies by taking n = 1: if ab \in P and a \notin P, then b \in P.^[27] Conversely, a primary ideal Q whose radical \sqrt{Q} is prime is called a primary ideal associated to that prime.^[28] For a module M over a commutative ring R, the associated primes of M, denoted \mathrm{Ass}(M), are the prime ideals \mathfrak{p} such that \mathfrak{p} = \mathrm{Ann}(m) for some nonzero m \in M.^[29] For an ideal I \subseteq R, the associated primes are those of the quotient module R/I, consisting of the minimal primes over I together with any embedded primes arising in decompositions.^[30] In Noetherian commutative rings, Krull's primary decomposition theorem states that every ideal admits a finite primary decomposition I = Q_1 \cap \cdots \cap Q_k, where each Q_i is primary.^[31] This decomposition is unique up to reordering when irredundant, meaning no Q_i contains the intersection of the others; the radicals \sqrt{Q_i} are then precisely the associated primes of I, with minimal such primes corresponding to the irreducible components and embedded primes indicating higher-dimensional overlaps.^[32] In noncommutative rings, primary ideals are defined analogously via left or right versions (e.g., an ideal U is left primary if for ideals I, J with IJ \subseteq U, either I \subseteq U or J \subseteq \mathrm{rad}(R/U)), but they play a less central role compared to the commutative case, as primary decompositions do not always exist for arbitrary ideals.^[33] However, in noncommutative Artinian rings, an analogous theory emerges through the Artin-Wedderburn theorem, decomposing the ring into a finite direct sum of simple Artinian rings and matrix rings over division rings, where primitive ideals (closely related to primaries) serve as building blocks.^[34]

Role in ring spectra

In commutative algebra, the prime spectrum of a ring R, denoted \operatorname{Spec}(R), consists of the set of all prime ideals of R endowed with the Zariski topology. In this topology, the closed subsets are the sets V(I) = \{ P \in \operatorname{Spec}(R) \mid I \subseteq P \} for arbitrary ideals I of R, and the open sets are complements of finite unions of such closed sets.^[35] This structure transforms \operatorname{Spec}(R) into a topological space that captures essential geometric and algebraic properties of R.^[36] The Zariski topology makes \operatorname{Spec}(R) a spectral space: a sober topological space (every irreducible closed subset has a unique generic point) admitting a basis of quasi-compact open sets closed under finite intersections.^[37] If R is Noetherian, then \operatorname{Spec}(R) also has a Noetherian topology, meaning every descending chain of closed sets stabilizes.^[38] In this space, each prime ideal P serves as the generic point of its closure \{ Q \in \operatorname{Spec}(R) \mid Q \supseteq P \}, which is the unique irreducible closed set containing it; this closure represents the "variety" associated to P.^[36] The topology unifies the study of prime ideals by providing a framework where algebraic inclusions correspond to spatial containments. The Krull dimension of R is defined topologically as the supremum of the lengths of strictly ascending chains of prime ideals in \operatorname{Spec}(R), measuring the "height" of the space.^[39] For example, in a polynomial ring k[x_1, \dots, x_n] over a field k, the longest such chain has length n, reflecting the affine dimension.^[39] Hilbert's Nullstellensatz establishes a profound link between \operatorname{Spec}(R) and classical algebraic geometry over algebraically closed fields: for R = k[x_1, \dots, x_n] with k algebraically closed, the maximal ideals correspond bijectively to points in affine n-space k^n, and more generally, the points of \operatorname{Spec}(R) parametrize the prime ideals containing the vanishing ideal of subvarieties.^[40] In noncommutative rings, an analog arises via the primitive spectrum \operatorname{Prim}(R), the set of primitive ideals (annihilators of simple left R-modules), equipped with the hull-kernel topology where closed sets are \{ P \in \operatorname{Prim}(R) \mid J \subseteq P \} for ideals J.^[41] This space serves as a noncommutative counterpart to \operatorname{Spec}(R), but its topology is generally coarser and lacks the full sobriety or Noetherian properties prevalent in the commutative case, limiting its geometric interpretability.^[41]

References

[1]
Prime Ideal -- from Wolfram MathWorld
A prime ideal is an ideal I such that if ab in I, then either a in I or b in I. For example, in the integers, the ideal a=<p> (ie, the multiples of p) is prime.
[2]
https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/An_Inquiry-Based_Approach_to_Abstract_Algebra_(Ernst)/08%3A_An_Introduction_to_Rings/8.04%3A_Maximal_and_Prime_Ideals
[3]
[PDF] 3. Prime and maximal ideals 3.1. Definitions and Examples ...
Definition. An ideal P in a ring A is called prime if P 6= A and if for every pair x, y of elements in A\P we have xy 6∈ P. Equivalently, if for every pair of ...
[4]
prime ideal in nLab
Jun 13, 2025 · Prime ideals are supposed to be a generalization of prime numbers from elements of the ring of integers to ideals in the sense of 'ideal elements' of an ...
[5]
[PDF] NOTES ON IDEALS 1. Introduction Let R be a commutative ring ...
Definition 6.1. An ideal I ⊂ R is called a prime ideal if the quotient ring R/I is an integral domain. We call I a maximal ideal if the quotient ring R/I is a ...
[6]
Prime Ideals in General Rings - jstor
Introduction. The concept of prime ideal has played an important role in the theory of commutative rings, but has not been used so extensively.
[7]
Ring Theory - MacTutor History of Mathematics
... prime ideals for non-commutative rings were not studied until 1949 by McCoy. In the 1940's attempts were made to prove results of the Wedderburn-Artin type ...
[8]
[PDF] 2 Localization and Dedekind domains - MIT Mathematics
Sep 13, 2021 · An important special case of localization occurs when p is a prime ideal in an integral domain A, and S = A − p (the complement of the set ...
[9]
[PDF] Commutative Algebra - MIT
Prime Ideals. 2. Prime Ideals. Prime ideals are the key to the structure of commutative rings. So we review the basic theory. Specifically, we define prime ...<|control11|><|separator|>
[10]
[PDF] FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 1 - Mathematics
Sep 26, 2005 · closed subsets of Spec R and prime ideals of R. From this conclude that in Spec R there is a bijection between points of Spec R and ...
[11]
Section 10.34 (00FS): Hilbert Nullstellensatz—The Stacks project
The first part (1) of Theorem 10.34.1 is called by many as Zeriski's Lemma, which is the H_3 in Zriski's 1947 paper "A new proof of Hilbert's Nullstellensatz" ...
[12]
Section 26.11 (01IR): Zariski topology of schemes—The Stacks project
Let X be a scheme. Any irreducible closed subset of X has a unique generic point. In other words, X is a sober topological space.
[13]
Prime Ideals (Chapter 3) - An Introduction to Noncommutative ...
We recall that a proper ideal P in a commutative ring R is prime if, whenever we have two elements a and b of R such that ab ∈ P, it follows that a ∈ P or b ∈ P ...
[14]
[PDF] 18.706 (S23), Full Lecture Notes - MIT OpenCourseWare
... simple module of a simple ring 𝑅 has to be faithful, because ker(𝑅 → End(𝐿)) is a 2-sided ideal in 𝑅, hence 0. 15. Page 16. Example 5.4: Primitive rings ...
[15]
[PDF] Lecture 22: Goldie Theorem, PI Rings - MIT OpenCourseWare
For example, every commutative domain where every annihilator of a nonzero element is zero and every nonzero ideal is essential is a right Goldie ring but not.
[16]
Structure of Noncommutative Rings - Northern Illinois University
The ring R is called a simple ring if (0) is a maximal ideal; it is called a prime ring if (0) is a prime ideal, and a semiprime ring if (0) is a semiprime ...
[17]
[PDF] Lecture 20: Ore Localization, Goldie Theorem - MIT OpenCourseWare
Definition 20.7: A ring 𝑅 is an Ore domain if it's a domain and 𝑅 \{0} satisfies O1. In this case, 𝑅𝑆 for 𝑆 = 𝑅 \{0} is clearly a skew field and 𝑅𝑆 = Frac(𝑅 ).<|separator|>
[18]
[PDF] Noncommutative localization in algebra and topology
non-zerodivisors of the ring R. The simplest extension to noncommutative rings is when the ring R satisfies the right Ore condition, that is given r ∈ R.
[19]
[PDF] Chapter 3 Prime ideals and Spec
The theory of prime ideals follows the commutative theory quite closely with some variations because our rings need not be commutative.
[20]
[PDF] Symbolic Powers - University of Utah Math Dept.
May 7, 2018 · An ideal Q in a ring R is called primary if the following holds for all a, b ∈ R: if ab ∈ Q, then a ∈ Q or bn ∈ Q for some n ⩾ 1. Remark 1.2.
[21]
[PDF] MA 252 notes: Commutative algebra - Brown Math
Feb 11, 2017 · This is the same as saying: in the nonzero ring A/q every zero-divisor is nilpotent. The inverse image of a primary ideal is a primary ideal.
[22]
[PDF] Commutative Algebra Chapter 4: Primary Decomposition
Definition. If p is a prime ideal, then any primary ideal q such that r(q) = p is called p-primary. Remark. Warning! If a is an ideal whose radical r(a) is ...
[23]
[PDF] Irena Swanson Primary decompositions - Purdue Math
For example, let R = k[X, Y ] be a polynomial ring in variables. X and Y over a field k. Let I be the ideal (X2,XY ). Then pI = (X) is a prime ideal in R.<|control11|><|separator|>
[24]
Section 10.63 (00L9): Associated primes—The Stacks project
A prime \mathfrak p of R is associated to M if there exists an element m \in M whose annihilator is \mathfrak p. The set of all such primes is denoted \text{Ass}_ ...
[25]
[PDF] Associated Primes - UC Berkeley math
A prime ideal P is associated to E if there is some element x of E such that Ann(x) = P. The set of such prime ideals is denoted by Ass(E). Thus ...
[26]
[PDF] 4.1 Primary Decompositions
Theorem: The radical of a primary ideal is the smallest prime ideal containing it. Proof: Let Q < A be primary. By an early result, the radical of Q , r(Q) = ...
[27]
[PDF] 4. Primary decomposition - Brandeis
p = r(a : x) for some x ∈ A. Theorem 4.10. If a = Pqi is a minimal primary decomposition then the prime ideals pi = r(qi) are exactly the associated primes for ...
[28]
[PDF] Primary Decomposition for Noncommutative Rings
Every principal left ideal ring is the direct product of a finite number of rings S in which P(S) is a prime ideal. Proof. We may assume that the ring ...
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PROPERTIES OF PRIMARY NONCOMMUTATIVE RINGS
If R is a completely N primary ring certainly A is a two sided ideal which is a maximal right ideal of R. Hence N = P = J. In this case suppose ab=0,
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[PDF] an introduction to the zariski topology - UChicago Math
This topological space, called the Zariski topology, gives a geometric way to interpret the algebra of a ring using the language of topology.
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[PDF] 1 The Zariski prime spectrum 2 Distinguished open subsets
We start with the underlying topological space. Let Spec(R) be the set of prime ideals of. R, i.e., the set of ideals p ⊆ R such that R/p is an integral ...
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[PDF] prime ideal structure in commutative rings(¹)
4. M. Hochster, Prime ideal structure in commutative rings, Thesis, Princeton Univ., Prince- ton, N. J., 1967.
[33]
[PDF] Krull Dimension
chain of prime ideals. In fact, one can give a purely algebraic definition of the Krull dimension of a ring. Inductive definition of dimension of spectral ...
[34]
[2309.14024] Early proofs of Hilbert's Nullstellensatz - arXiv
Sep 25, 2023 · Access Paper: View a PDF of the paper titled Early proofs of Hilbert's Nullstellensatz, by Jan Stevens. View PDF · TeX Source · Other Formats.
[35]
[PDF] An Introduction to Noncommutative Spaces and their Geometry - arXiv
... primitive ideal space is a given (finitary, noncommutative) lattice P. We shall briefly describe this procedure while referring to. [44, 45] for more details ...