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Minimal prime ideal

In commutative algebra, a minimal prime ideal of a commutative ring R is a prime ideal p such that no other prime ideal of R is properly contained in p.^[1] Every nonzero commutative ring possesses at least one minimal prime ideal, which can be viewed as a minimal element in the partially ordered set of prime ideals under inclusion.^[1] Minimal prime ideals play a central role in the structure theory of rings, particularly through their correspondence with the irreducible components of the spectrum \operatorname{Spec}(R), the set of all prime ideals equipped with the Zariski topology. In a Noetherian ring, the minimal primes over any ideal I are finite in number and are contained in every prime ideal containing I, ensuring a well-defined decomposition of the spectrum into irreducible closed subsets.^[2] For instance, each minimal prime p defines an irreducible closed set V(p) that is maximal among irreducible subsets of \operatorname{Spec}(R), linking algebraic properties to geometric intuition. In reduced rings—those without nonzero nilpotent elements—the union of all minimal prime ideals precisely equals the set of zero-divisors, highlighting their connection to the ring's integral structure.^[3] For a reduced Noetherian ring, localization at a minimal prime p yields a field, underscoring the "generic" nature of these ideals.^[4] In integral domains, the zero ideal is the unique minimal prime, while in more general settings, such as geometrically irreducible algebras over a field k, the property of having a unique minimal prime persists under base change to field extensions k'/k.^[4]^[5] These ideals also feature prominently in advanced topics, including the associated primes of modules and criteria for normality in Noetherian rings, where conditions like Serre's (R1) and (S2) relate minimal primes to regularity at height-zero primes.^[6]^[7]

Definition and Characterization

Formal Definition

In a commutative ring R with identity, an ideal P is prime if whenever ab \in P for a, b \in R, then a \in P or b \in P.^[8] This condition ensures that the quotient ring R/P is an integral domain.^[8] A prime ideal P of R is minimal if there is no prime ideal Q of R such that Q \subsetneq P.^[9] Equivalently, P contains no other prime ideal of R strictly contained within it.^[8] The set of all prime ideals of R, denoted \operatorname{Spec}(R), forms a partially ordered set under inclusion, and the minimal prime ideals are precisely the minimal elements of this poset.^[8]

Equivalent Characterizations

A prime ideal \mathfrak{p} of a commutative ring R containing an ideal I is minimal over I if there is no prime ideal \mathfrak{q} such that I \subsetneq \mathfrak{q} \subsetneq \mathfrak{p}. The set of all minimal prime ideals over I, denoted \mathrm{Min}(I), consists precisely of those prime ideals \mathfrak{p} \supset I that are minimal with respect to inclusion among all primes containing I:

\mathrm{Min}(I) = \{ \mathfrak{p} \in \mathrm{Spec}(R) \mid \mathfrak{p} \supset I \text{ and there is no } \mathfrak{q} \in \mathrm{Spec}(R) \text{ with } I \subsetneq \mathfrak{q} \subsetneq \mathfrak{p} \}.

This set is nonempty for any proper ideal I in a nonzero ring R, as shown below using Zorn's lemma.^[10] The existence of minimal primes over any ideal follows from Zorn's lemma applied to the partially ordered set of prime ideals containing I, ordered by inclusion. Consider the collection \mathcal{S} of all prime ideals of R that contain I, partially ordered by inclusion. This set is nonempty because I is proper, so there exists a maximal ideal containing I (by Zorn's lemma on the poset of ideals containing I). Any chain \{ \mathfrak{p}_\alpha \} in \mathcal{S} has a lower bound given by the intersection \bigcap \mathfrak{p}_\alpha, which contains I. This intersection is prime: suppose xy \in \bigcap \mathfrak{p}_\alpha but x \notin \bigcap \mathfrak{p}_\alpha. Then there exists \mathfrak{p}_0 in the chain with x \notin \mathfrak{p}_0. The subcollection \{ \mathfrak{p}_\alpha \mid \mathfrak{p}_\alpha \supset \mathfrak{p}_0 \} is a subchain whose intersection contains xy; since x \notin \mathfrak{p}_0 \subset \mathfrak{p}_\alpha for all such \mathfrak{p}_\alpha, if y \notin \bigcap \{ \mathfrak{p}_\alpha \mid \mathfrak{p}_\alpha \supset \mathfrak{p}_0 \}, there would be some \mathfrak{p}_1 \supset \mathfrak{p}_0 with y \notin \mathfrak{p}_1, but then xy \in \mathfrak{p}_1 and x \notin \mathfrak{p}_1 imply a contradiction to \mathfrak{p}_1 prime. Thus y is in the intersection, so the full intersection is prime. By Zorn's lemma, \mathcal{S} has minimal elements, which are precisely the minimal primes over I. Every prime ideal of R contains a minimal prime (over the zero ideal).^[10] An equivalent characterization arises from localization: a prime ideal \mathfrak{p} of R is minimal (over the zero ideal, or more generally over any I \subset \mathfrak{p}) if and only if in the localization R_\mathfrak{p}, every element of the maximal ideal \mathfrak{p} R_\mathfrak{p} is nilpotent. To see this, note that the prime ideals of R_\mathfrak{p} correspond bijectively to the prime ideals of R contained in \mathfrak{p}. Thus, \mathfrak{p} minimal implies \mathrm{Spec}(R_\mathfrak{p}) consists solely of the maximal ideal \mathfrak{m} = \mathfrak{p} R_\mathfrak{p}. If some x \in \mathfrak{m} is not nilpotent, then the principal open D(x) in \mathrm{Spec}(R_\mathfrak{p}) is nonempty, since non-nilpotent elements are not contained in every prime ideal. This yields a prime not containing x, hence distinct from \mathfrak{m}, a contradiction. Conversely, if every element of \mathfrak{m} is nilpotent, any proper prime of R_\mathfrak{p} would be contained in the nilradical (intersection of all primes, hence \mathfrak{m}), but the nilradical equals \mathfrak{m} only if no smaller primes exist. If R is reduced (nilradical zero), this further implies \mathfrak{m} = 0, so R_\mathfrak{p} is a field.^[4]

Basic Properties

Minimal Primes over Ideals

In a commutative ring R with identity, given an ideal I \subseteq R, a prime ideal \mathfrak{p} \subseteq R is said to be minimal over I if \mathfrak{p} \supseteq I and there exists no prime ideal \mathfrak{q} such that I \subseteq \mathfrak{q} \subsetneq \mathfrak{p}.^[11] These minimal primes over I represent the "smallest" prime ideals containing I in the partially ordered set of prime ideals ordered by inclusion. The existence of at least one minimal prime ideal over any ideal I follows from Zorn's lemma applied to the collection of prime ideals containing I, which is nonempty (as R has prime ideals by standard results in ring theory) and inductive under inclusion.^[11] Thus, every proper ideal in R is contained in some minimal prime ideal over it. Moreover, the radical of I, denoted \sqrt{I} = \{ r \in R \mid r^n \in I \text{ for some } n \geq 1 \}, equals the intersection of all prime ideals containing I, which coincides with the intersection of the minimal prime ideals over I since every prime containing I contains a minimal one.^[11] In Noetherian rings, the set of minimal primes over any ideal I is finite; this follows from the primary decomposition theorem, where the minimal primes correspond to the isolated components in the decomposition of I.^[11] This finiteness is crucial for computational aspects and structural theorems in commutative algebra, such as those involving the support of modules.

Intersection and Radical

In commutative ring theory, the collection of all minimal prime ideals of a ring R plays a fundamental role in describing the nilpotent elements of R. Specifically, the intersection of all minimal prime ideals of R is the nilradical of R, denoted \Nil(R).^[12] The nilradical \Nil(R) consists of all nilpotent elements of R, i.e., elements x \in R such that x^n = 0 for some positive integer n. It coincides with the radical of the zero ideal, \sqrt{0}, and satisfies the equality

\sqrt{0} = \Nil(R) = \bigcap \{ P \mid P \text{ is a minimal prime ideal of } R \}.

This relation follows from the fact that \Nil(R) is also the intersection of all prime ideals of R, and every prime ideal properly contains at least one minimal prime ideal, ensuring the intersections coincide.^[13]^[14] In any commutative ring R, the minimal prime ideals are precisely the prime ideals containing \Nil(R) that are minimal with respect to inclusion; in Noetherian rings, these correspond to the isolated primary components in the primary decomposition of the zero ideal, providing a unique decomposition of \Nil(R) into primary ideals associated to these minimal primes.^[12] As a generalization, the radical of an arbitrary ideal I \subseteq R is the intersection of all prime ideals containing I, with the minimal such primes over I determining the isolated components analogous to the case I = 0 in Noetherian rings.^[13]

Examples

In Polynomial Rings

In the polynomial ring k[x, y] over a field k, the zero ideal is the unique minimal prime ideal, since k[x, y] is an integral domain. The principal ideals (x) and (y) are prime ideals generated by irreducible elements in this unique factorization domain.^[15] In contrast, the ideal (x, y) is a prime ideal—being maximal in this two-dimensional ring—but it is not minimal, since it properly contains the smaller prime (x), for example.^[15] A concrete computation illustrates minimal primes over a non-prime ideal: consider I = (xy) in k[x, y]. The minimal prime ideals containing I are precisely (x) and (y), as any prime ideal \mathfrak{p} \supseteq I must contain either x or y (since \mathfrak{p} is prime), and these are the smallest such primes.^[16] Moreover, the radical \sqrt{I} equals the intersection (x) \cap (y), which coincides with I itself in this case, confirming the primary decomposition I = (x) \cap (y).^[16] Geometrically, in polynomial rings over algebraically closed fields, the minimal prime ideals over an ideal I correspond bijectively to the irreducible components of the affine variety V(I) defined by the zero set of I in affine space.^[17] For instance, the variety V((xy)) consists of the union of the x-axis and y-axis, whose irreducible components are captured by the minimal primes (x) and (y).^[16]

In Quotient Rings

In commutative algebra, the prime ideals of a quotient ring R/I, where R is a commutative ring and I is an ideal of R, are in bijective correspondence with the prime ideals of R that contain I. Specifically, this bijection is given by mapping a prime ideal \mathfrak{q} of R/I to its preimage \mathfrak{p} = \pi^{-1}(\mathfrak{q}) under the canonical surjection \pi: R \to R/I, where \mathfrak{p} contains I, and conversely by mapping a prime \mathfrak{p} \supseteq I to \mathfrak{p}/I.^[18] This correspondence preserves inclusions in the reverse direction: if \mathfrak{q}_1 \subseteq \mathfrak{q}_2 in \operatorname{Spec}(R/I), then \pi^{-1}(\mathfrak{q}_1) \subseteq \pi^{-1}(\mathfrak{q}_2) in \operatorname{Spec}(R).^[18] Consequently, the minimal prime ideals of R/I correspond precisely to the minimal prime ideals of R that contain I, often called the minimal primes over I. A prime ideal \mathfrak{p}/I is minimal in \operatorname{Spec}(R/I) if and only if there is no prime ideal of R strictly between I and \mathfrak{p}.^[18] This descent property highlights how quotienting by I "lifts" the minimal structure from the primes over I in the original ring. A concrete computation illustrates this in the ring of integers: consider R = \mathbb{Z} and I = (4), so R/I \cong \mathbb{Z}/4\mathbb{Z}. The prime ideals of \mathbb{Z} containing (4) are those generated by primes dividing 4, namely (2), which is minimal over (4). Thus, the unique minimal prime ideal of \mathbb{Z}/4\mathbb{Z} is (2)/(4) \cong \{0, 2\} \mod 4, and \mathbb{Z}/4\mathbb{Z} / ((2)/(4)) \cong \mathbb{Z}/2\mathbb{Z} is an integral domain.^[18] More generally, the spectrum \operatorname{Spec}(R/I) embeds homeomorphically into \operatorname{Spec}(R) as the closed subset V(I) consisting of all primes containing I. Under this embedding, minimality is preserved: the minimal elements of \operatorname{Spec}(R/I) map to the minimal elements of V(I).^[18] This topological perspective underscores the role of quotient rings in studying the geometry of ideals via the Zariski topology.

Geometric and Dimensional Aspects

Associated Varieties

In algebraic geometry, minimal prime ideals over an ideal I in the polynomial ring k[x_1, \dots, x_n], where k is an algebraically closed field, provide a geometric interpretation through their correspondence to the affine variety V(I) \subset \mathbb{A}^n. Specifically, the minimal primes P_1, \dots, P_r over I are in bijection with the irreducible components of V(I), where each component is the variety V(P_i) associated to a minimal prime P_i. This decomposition reflects the fact that V(I) can be uniquely expressed as a finite union of these irreducible subvarieties V(P_i), with no proper containment among them.^[19]^[20] The radical of the ideal \sqrt{I} plays a central role in this correspondence, as it equals the intersection of all minimal primes over I: \sqrt{I} = P_1 \cap \cdots \cap P_r. Geometrically, this implies that V(\sqrt{I}) is the union of the irreducible varieties V(P_i) for the minimal primes P_i, and since \sqrt{I} determines the same variety as I by the properties of the Zariski topology, the variety V(I) inherits this decomposition into its irreducible components. This structure ensures that the minimal primes capture the "essential" geometric structure of V(I) without embedded components.^[19]^[17] Hilbert's Nullstellensatz strengthens this connection by establishing a precise dictionary between ideals and varieties over algebraically closed fields. In particular, the minimal prime ideals over I arise as the kernels of surjective homomorphisms from k[x_1, \dots, x_n] onto the coordinate rings of the irreducible varieties V(P_i), where each such coordinate ring k[x_1, \dots, x_n]/P_i is an integral domain reflecting the irreducibility of V(P_i). This implication of the Nullstellensatz underscores how minimal primes encode the irreducible geometric building blocks of affine varieties.^[21]^[20]

Height and Krull Dimension

The height of a minimal prime ideal P in a commutative ring R is zero, as there are no prime ideals of R strictly contained in P. Equivalently, the height \ht(P) is the Krull dimension of the localization R_P, and \dim(R_P) = 0 because \Spec(R_P) consists solely of the single prime ideal P R_P, admitting no strict chains of primes.^[22] In general, the height \ht(Q) of any prime ideal Q in R is the supremum of the lengths n of descending chains of distinct prime ideals Q \supset Q_1 \supset \cdots \supset Q_n. For a minimal prime P, this supremum is zero, measuring the absence of primes below P and thus the "codimension zero" status of the corresponding irreducible component in algebraic geometry interpretations. The Krull dimension \dim(R) of R is then the supremum of these heights over all prime ideals (or equivalently over maximal ideals), which equals the supremum of lengths of ascending chains of primes starting from minimal primes. In terms of quotients, \dim(R) = \sup \{ \dim(R/P) \mid P \text{ a minimal prime ideal of } R \}, where each \dim(R/P) gives the dimension of the integral domain R/P associated to the irreducible component V(P).^[23]^[24] In catenary rings, where all saturated chains of prime ideals between a given minimal prime and a maximal ideal containing it have the same length, the relation \ht(Q) + \dim(R/Q) = \dim(R) holds for every prime ideal Q. For minimal primes P, with \ht(P) = 0, this simplifies to \dim(R/P) = \dim(R). A Noetherian ring R is equidimensional if \dim(R/P) = \dim(R) for every minimal prime P, ensuring all irreducible components have the same dimension with no embedded components of lower dimension.^[25]^[26]

Advanced Relations

To Associated Primes

In commutative algebra, for a commutative ring R and an R-module M, the associated primes of M, denoted \mathrm{Ass}_R(M), are the prime ideals \mathfrak{p} \subset R such that there exists an injective R-module homomorphism R/\mathfrak{p} \hookrightarrow M.^[27] Equivalently, \mathfrak{p} \in \mathrm{Ass}_R(M) if there is a nonzero element m \in M with annihilator \mathrm{Ann}_R(m) = \mathfrak{p}.^[27] The set \mathrm{Ass}_R(M) captures the prime ideals arising as annihilators of cyclic submodules of M, providing key information about the prime ideals "embedded" in the structure of M.^[28] The minimal elements of \mathrm{Ass}_R(M) (with respect to inclusion) are precisely the minimal prime ideals of R containing the annihilator ideal \mathrm{Ann}_R(M).^[27] These minimal associated primes correspond to the isolated primary components in decompositions of M, while non-minimal (embedded) associated primes reflect deeper torsion or zero-divisor structures.^[27] For instance, if I \subset R is an ideal, the minimal primes over I are the minimal associated primes of the quotient module R/I.^[27] When M = R, the associated primes \mathrm{Ass}_R(R) consist of all primes \mathfrak{p} such that R/\mathfrak{p} embeds into R, and every minimal prime of R belongs to \mathrm{Ass}_R(R).^[27] Thus, the minimal elements of \mathrm{Ass}_R(R) are exactly the minimal primes of R, though \mathrm{Ass}_R(R) may include additional embedded primes in non-reduced cases.^[27] If R is Noetherian and M is a finitely generated R-module, then \mathrm{Ass}_R(M) is a finite set.^[27] This finiteness ensures that the associated primes, including their minimal elements, form a controlled collection that fully describes the prime support of M.^[27]

In Primary Decomposition

In a Noetherian ring R, the primary decomposition theorem asserts that every ideal I \subseteq R admits a primary decomposition I = \bigcap_{i=1}^n Q_i, where each Q_i is a primary ideal with radical \sqrt{Q_i} = P_i a prime ideal.^[29] The primes P_i appearing in such a decomposition are precisely the associated primes of I, and among these, the minimal primes over I—those P minimal with respect to inclusion among the P_i—play a distinguished role as the radicals of the isolated primary components.^[29] This theorem, originally established by Lasker for polynomial rings and generalized by Noether to arbitrary Noetherian rings, underpins the structural analysis of ideals in commutative algebra.^[30]^[31] A primary decomposition is irredundant if the primes P_i are distinct and no single Q_i contains the intersection of the remaining components, ensuring that each term is essential.^[29] In any irredundant primary decomposition of I, the minimal primes over I uniquely determine the isolated primary components, which are the Q_P such that \sqrt{Q_P} = P is minimal over I; these components are independent of the choice of decomposition.^[29] The embedded primary components, corresponding to non-minimal primes that properly contain some minimal prime over I, are not uniquely determined but can be isolated in the decomposition. Consequently, every primary decomposition of I refines to a form I = \left( \bigcap_{P \text{ minimal over } I} Q_P \right) \cap J, where each Q_P is P-primary and J is the intersection of the embedded primary components (possibly empty).^[29] This separation highlights the minimal primes as the "essential" structure underlying the ideal, with embedded components capturing additional complexity.^[29]

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