Set Of Prime Ideals Research Articles

The prime spectrum of an 𝐿-algebra

We prove that the lattice of ideals of an arbitrary L L -algebra is distributive. As a consequence, a spectral theory applies with no restriction. We also study the spectrum (i.e. the set of prime ideals) of L L -algebras and characterize prime ideals in topological terms.

On the topological structure of the (non-)finitely generated locus of Frobenius algebras emerging from Stanley–Reisner rings

Let U be the set of prime ideals P of the completion of a Stanley–Reisner ring S, such that the localization at P of the Frobenius algebra of the injective hull of the residue field of S is a finitely generated algebra. We give a partial answer to a conjecture made by M. Katzman about the openness of U. Specifically, we show that U has non-empty interior and we present some sufficient conditions for a principal open set D(f) to be contained in U, and for intersections of closed and principal opens to be contained in the complement of U.

On sums and products of primitive elements

ABSTRACTLet R be a (commutative integral) domain, with K its quotient field and R′ its integral closure (in K). Let 𝒫 be the set of elements u∈K such that u is primitive over R; i.e., such that u is the root of a polynomial over R having a unit coefficient. Then, 𝒫 is a ring (necessarily K) ⇔ 𝒫 is closed under products ⇔ R′ is a Prüfer domain. In general, 𝒫 is closed under powers. For u,v∈𝒫, necessary and sufficient conditions are given for u+v (resp., uv) to belong to 𝒫. Also, 𝒫 is used to characterize when R is a quasi-local integrally closed domain and when R is a pseudo-valuation domain. If R is quasi-local, each element of K is expressible as the sum of two (possibly equal) elements of 𝒫. The set of primitive elements is determined for lying-over pairs and for extensions of domains with the same sets of prime ideals. In this study of the 𝒫 construction, R and K are replaced, whenever possible, by an arbitrary commutative ring and its total quotient ring or, more generally, by any inclusion of commutative rings.

Integral domains with finitely many spectral semistar operations

Let D be a finite-dimensional integral domain, Spec(D) be the set of prime ideals of D, and SpSS(D) be the set of spectral semistar operations on D. Mimouni gave a complete description for the prime ideal structure of D with SpSS(D) = n + dim(D) for 1 ≤ n ≤ 5 except for the quasi-local cases of n = 4, 5. In this paper, we show that there is an integral domain D such that SpSS(D) = n+dim(D) for all positive integers n with n ≠ 2. As corollaries, we completely characterize the quasi-local domains D with SpSS(D) = n+dim(D) for n = 4, 5. Furthermore, we also present the lower and upper bounds of SpSS(D) when Spec(D) is a finite tree.

Ideal Structure of Leavitt Path Algebras with Coefficients in a Unital Commutative Ring

For a (countable) graph E and a unital commutative ring R, we analyze the ideal structure of the Leavitt path algebra L R (E) introduced by Mark Tomforde. We first modify the definition of basic ideals and then develop the ideal characterization of Mark Tomforde. We also give necessary and sufficient conditions for the primeness and the primitivity of L R (E), and we then determine prime graded basic ideals and left (or right) primitive graded ideals of L R (E). In particular, when E satisfies Condition (K) and R is a field, they imply that the set of prime ideals and the set of primitive ideals of L R (E) coincide.

LINK CLOSED SETS OF PRIME IDEALS AND STABILITY ON BIMODULES

Over a Noetherian ring R, each of a weakly symmetric pair of torsion radicals is shown to be stable on Noetherian R-R bimodules if and only if the set of prime ideals that are closed with respect to each torsion radical is closed under links. Such a pair for R, termed a weakly symmetric stable pair, is extended to a weakly symmetric stable pair for any Noetherian extension ring of R. In case classical Krull dimension is a link invariant, we give a positive answer to the incomparability question for linked prime ideals in certain extension rings of R.

Graphical properties of the bipartite graph of Spec(Z[x])\{0}

Consider $Spec(Z[x])$, the set of prime ideals of $Z[x]$ as a partially ordered set under inclusion. By removing the zero ideal, we denote $G_{Z}=Spec(Z[x])\{0}$ and view it as an infinite bipartite graph with the prime ideals as the vertices and the inclusion relations as the edges. In this paper, we investigate fundamental graph theoretic properties of $G_{Z}$. In particular, we describe the diameter, circumference, girth, radius, eccentricity, vertex and edge connectivity, and cliques of $G_{Z}$. The complement of $G_{Z}$ is investigated as well.

The set of prime divisors of generalized denominator ideals

Let R be a Noetherian domain with quotient field K. Let A be an integral domain which contains R and whose elements are algebraic over K. We define \({{\rm Eass}_{R}(A/R)}\) to be the set of prime ideals \({\mathfrak{p}}\) ’s of R such that \({\mathfrak{p}}\) is a prime divisor of a generalized denominator ideal I[β] for some \({\beta \in A}\). Assume that \({A = R[\alpha_{1}, \ldots,\alpha_{n}]}\). We investigate the relation between \({{\rm Eass}_{R}(A/R)}\) and \({\cup_{i = 1}^{n} {\rm Ass}_{R}(R/I_{[\alpha_{i}]})}\). Furthermore, for a finite subset Δ of \({{\rm Eass}_{R}(A/R)}\), we construct the subring A Δ of A such that A Δ is the largest among those B’s with Δ = Eass R ( B / R ) .

Quasi-finite modules and asymptotic prime divisors

Let A be a Noetherian ring, J⊆A an ideal and C a finitely generated A-module. In this note we would like to prove the following statement. Let {In}n⩾0 be a collection of ideals satisfying: (i) In⊇Jn, for all n, (ii) Js⋅Is⊆Ir+s, for all r,s⩾0 and (iii) In⊆Im, whenever m⩽n. Then AssA(InC/JnC) is independent of n, for n sufficiently large. Note that the set of prime ideals ⋃n⩾1AssA(InC/JnC) is finite, so the issue at hand is the realization that the primes in AssA(InC/JnC)do not behave periodically, as one might have expected, say if ⨁n⩾0In were a Noetherian A-algebra generated in degrees greater than one. We also give a multigraded version of our results.

Minimal Zero-Dimensional Extensions of Rings of Dimension Greater than One

Let R be a commutative ring with Noetherian spectrum in which zero is a primary ideal. We determine the minimal zero-dimensional extensions of R when every regular prime ideal of R is contained in only finitely many prime ideals. This extends previous results of the first author for dim (R) ≤1. We also present a characterization of the partially ordered set of prime ideals in a ring with Noetherian spectrum.

On Localizing Systems in a Prűfer Domain

The aim of this article is to describe all localizing systems (l.s.) of a Prufer domain by using the description of l.s. in a valuation ring (Theorem 1). By using the notion of localization of l.s. of a Prufer domain, we succeeded to determine all l.s. 𝒢 in a Prufer domain (Theorems 7 and 10) by using a suitable set of prime ideals {P(𝒢, M)} M∈MaxR . Finally, in Theorem 13, we describe the localization of a Prufer domain R with respect to a l.s. of R.

Integral Domains in Which Nonzero Locally Principal Ideals are Invertible

We study locally principal ideals and integral domains, called LPI domains, in which every nonzero locally principal ideal is invertible. We show that a finite character intersection of LPI overrings is an LPI domain. Hence if a domain D is a finite character intersection for some set of prime ideals of D, then D is an LPI domain.

Formal Fibers of Unique Factorization Domains

AbstractLet (T,M) be a complete local (Noetherian) ring such that dimT ≥ 2 and |T| = |T/M| and let ﹛pi﹜i∈𝒥 be a collection of elements of T indexed by a set I so that |𝒥| < |T|. For each i ∈ 𝒥, let Ci := ﹛Qi1, … ,Qini ﹜ be a set of nonmaximal prime ideals containing pi such that the Qi j are incomparable and pi ∈ Qjk if and only if i = j. We provide necessary and sufficient conditions so that T is the m -adic completion of a local unique factorization domain (A,m ), and for each i ∈ I, there exists a unit ti of T so that pi ti ∈ A andCi is the set of prime ideals Q of T that are maximal with respect to the condition that Q ∩ A = pi tiA.We then use this result to construct a (nonexcellent) unique factorization domain containingmany ideals for which tight closure and completion do not commute. As another application, we construct a unique factorization domain A most of whose formal fibers are geometrically regular.

Cauchon diagrams for quantized enveloping algebras

Let g be a finite-dimensional complex simple Lie algebra, K a commutative field and q a nonzero element of K which is not a root of unity. To each reduced decomposition of the longest element w 0 of the Weyl group W corresponds a PBW basis of the quantised enveloping algebra U q + ( g ) , and one can apply the theory of deleting-derivation to this iterated Ore extension. In particular, for each decomposition of w 0 , this theory constructs a bijection between the set of prime ideals in U q + ( g ) that are invariant under a natural torus action and certain combinatorial objects called Cauchon diagrams. In this paper, we give an algorithmic description of these Cauchon diagrams when the chosen reduced decomposition of w 0 corresponds to a good ordering (in the sense of Lusztig (1990) [Lus90]) of the set of positive roots. This algorithmic description is based on the constraints that are coming from Lusztig's admissible planes Lusztig (1990) [Lus90]: each admissible plane leads to a set of constraints that a diagram has to satisfy to be Cauchon. Moreover, we explicitly describe the set of Cauchon diagrams for explicit reduced decomposition of w 0 in each possible type. In any case, we check that the number of Cauchon diagrams is always equal to the cardinal of W. In Cauchon and Mériaux (2008) [CM08], we use these results to prove that Cauchon diagrams correspond canonically to the positive subexpressions of w 0 . So the results of this paper also give an algorithmic description of the positive subexpressions of any reduced decomposition of w 0 corresponding to a good ordering.

On the finiteness properties of Matlis duals of local cohomology modules

Let R be a complete semi-local ring with respect to the topology defined by its Jacobson radical, a an ideal of R, and M a finitely generated R-module. Let D R (−) := Hom R (−, E), where E is the injective hull of the direct sum of all simple R-modules. If n is a positive integer such that Ext R j (R/a, D R (H a t (M))) is finitely generated for all t > n and all j ⩾ 0, then we show that Hom R (R/a, D R (H a n (M))) is also finitely generated. Specially, the set of prime ideals in Coass R (H a n (M)) which contains a is finite.

Gorenstein graded rings associated to ideals

Let A be a Noetherian local ring with the maximal ideal m and dim A = d. Let I(6= A) be an ideal in A and s = htA I. This article studies the question of when the associated graded ring G(I) := ⊕ i≥0 I /I is Gorenstein. To state our result, we set up some notation. Let ` be an integer such that ` ≤ d and let J be a reduction of I generated by elements a1, a2, . . . , a`. We denote by rJ(I) the reduction number of I with respect to J . The analytic spread of I is λ(I) := dim A/m ⊗A G(I). Then s ≤ λ(I) ≤ `. We always assume that the generating set {a1, a2, . . . , a`} of J is a basic generating set for J in the sense of Aberbach, Huneke, and Trung [AHT], which means that JiAq is a reduction of IAq for all q ∈ V(I) with i = htA q < `. Here let V(I) be a set of prime ideals in A containing I and Ji := (a1, a2, . . . , ai) for 0 ≤ i ≤ `. By [AHT], 7.2, there always exists a basic generating set for J if the field A/m is infinite. Let

On pairs of subrings with a common set of proper ideals

A study of pairs of commutative rings with the same set of prime ideals appears in the literature. In this paper, we investigate pairs of subrings, not necessarily commutative, with a common set of proper ideals.

A Description of Ideals in Noetherian Rings

For any commutative ring R there is a theory T(R) in a propositional language whose models are exactly the ideals of R. These ideals form a closed subspace of the Tychonov product topology on 2 R and we introduce an ultraproduct of sets operation on 2 R which allows us to “compute” limits in this space. The property that R is Noetherian is equivalent to the condition that any ultraproduct of its ideals is equal to an intersection of some of these ideals over some ultrafilter set. For any countable Noetherian ring R,this topological space is homeomorphic to a countable successor ordinal with its interval topology. We describe how the ideals of a Noetherian ring R are related to each other in terms of limits in this space as well as infinite intersections. Another description of these ideals is given in terms of their complete theories and their Cantor rank. This method works for any extension of T(R) and so there is a similar analysis for the set of prime ideals of R. Any non-empty closed set in the space Spec(R) has finitely many minimal elements of maximum Cantor rank. For countable Noetherian R of finite Krull dimension k ≥ 1,Spec(R) is homeomorphic to either ω k · j + 1 or ω k−1 · j + 1 for some j < ω. For the prime ideals of Z[x 1,…,x k ] an explicit representation of each ideal of Krull rank 1 is given as a limit or as intersections of its ideals of Krull rank 2.

The Extensional Structure of Commutative Noetherian Rings

We show that finitely generated modules over a commutative Noetherian ring can be classified, up to isomorphism of submodule series, in a manner analogous to the classification of integers as products of prime numbers. In outline, two such modules have isomorphic submodule series if and only if 1) the set of minimal associated prime ideals of these modules coincide, 2) the multiplicities of these modules at these prime ideals coincide, and 3) the modules represent the same element in a certain group corresponding to the above set of prime ideals. Regarding condition 3), we show that, in the very special case that the ring is a Dedekind domain, the group corresponding to the prime ideal (0) is the ideal class group of the ring.

Oversemigroups of a valuation semigroup

By a semigroup we mean a submonoid of a torsion-free group in this paper. We investigate some behaviour of prime ideals among semigroups and their oversemigroups. In particular, we consider a oversemigroup T of a valuation semigroup S such that the set of prime ideals P of S satisfying the property that P+T⊂≠T equals to Spec(T). Also, we study some conditions that a valuation semigroup is completely integrally closed. We give some examples of a valuation semigroup of dimension two and a completely integrally closed semigroup.