Distinct Prime Ideals Research Articles

On normal modules

Recall that a commutative ring is said to be a normal ring if it is reduced and every two distinct minimal prime ideals are comaximal. A finitely generated reduced R-module is said to be a normal module if every two distinct minimal prime submodules are comaximal. The concepts of normal modules and locally torsion free modules are different, whereas they are equal in theory of commutative rings. We give many properties and examples of normal modules, we use them to characterize locally torsion free modules and Baer modules. Also, we give the topological characterizations of normal modules.

Products and intersections of prime-power ideals in Leavitt path algebras

We continue a very fruitful line of inquiry into the multiplicative ideal theory of an arbitrary Leavitt path algebra [Formula: see text]. Specifically, we show that factorizations of an ideal in [Formula: see text] into irredundant products or intersections of finitely many prime-power ideals are unique, provided that the ideals involved are powers of distinct prime ideals. We also characterize the completely irreducible ideals in [Formula: see text], which turn out to be prime-power ideals of a special type, as well as ideals that can be factored into products or intersections of finitely many completely irreducible ideals.

A prime analogue Erdős-Pomerance result for Drinfeld modules with arbitrary endomorphism rings

Let k \mathbf {k} be a global function field, and let A \mathbf {A} be the elements of k \mathbf {k} regular outside a fixed place ∞ \infty . Let ϕ : A → K { τ } \phi :\mathbf {A}\to K\{\tau \} be a Drinfeld module of generic characteristic and rank n n . For a prime ℘ \wp of K K of good reduction, let F ℘ \mathbb {F}_\wp be the residue field at ℘ \wp , and let χ A ( ϕ ( F ℘ ) ) \chi _\mathbf {A}\big (\phi (\mathbb {F}_\wp )\big ) be the Euler-Poincaré characteristic of F ℘ \mathbb {F}_\wp viewed as an A \mathbf {A} -module. We determine the normal order of the number of distinct prime ideals of A \mathbf {A} dividing χ A ( ϕ ( F ℘ ) ) \chi _\mathbf {A}\big (\phi (\mathbb {F}_\wp )\big ) , denoted by ω A ( χ A ( ϕ ( F ℘ ) ) ) \omega _\mathbf {A}\Big (\chi _\mathbf {A}\big (\phi (\mathbb {F}_\wp )\big )\Big ) , as ℘ \wp runs over primes of K K of degree x x with a specified splitting behaviour. Furthermore, let a ∈ K a\in K be non-torsion for ϕ \phi , and let f a ( ℘ ) f_a(\wp ) be the Euler-Poincaré characteristic of the submodule of ϕ ( F ℘ ) \phi (\mathbb {F}_\wp ) generated by a a modulo ℘ \wp . We also consider the problem of determining the distribution of ω A ( f a ( ℘ ) ) \omega _\mathbf {A}\big (f_a(\wp ) \big ) as ℘ \wp runs over primes of K K of degree x x with a specified splitting behaviour. Note that we do not make restrictions on A \mathbf {A} , ϕ \phi , or its endomorphism ring E n d K {sep} ( ϕ ) \rm {End}_{K^\textrm {{sep}}}(\phi ) .

The Erdős–Burgess constant of the multiplicative semigroup of a factor ring of 𝔽q[x

Let [Formula: see text] be a commutative semigroup endowed with a binary associative operation [Formula: see text]. An element [Formula: see text] of [Formula: see text] is said to be idempotent if [Formula: see text]. The Erdős–Burgess constant of [Formula: see text] is defined as the smallest [Formula: see text] such that any sequence [Formula: see text] of terms from [Formula: see text] and of length [Formula: see text] contains a nonempty subsequence, the sum of whose terms is idempotent. Let [Formula: see text] be a prime power, and let [Formula: see text] be the polynomial ring over the finite field [Formula: see text]. Let [Formula: see text] be a quotient ring of [Formula: see text] modulo any ideal [Formula: see text]. We gave a sharp lower bound of the Erdős–Burgess constant of the multiplicative semigroup of the ring [Formula: see text], in particular, we determined the Erdős–Burgess constant in the case when [Formula: see text] is the power of a prime ideal or a product of pairwise distinct prime ideals in [Formula: see text].

On the gap between prime ideals

We define a gap function to measure the difference of two distinct prime ideals in a given number field. In this paper, we determine all quadratic fields and cyclotomic fields satisfying the condition: there exist two distinct prime ideals whose gap is 1.

Coprime Sensing via Chinese Remaindering Over Quadratic Fields—Part I: Array Designs

A coprime antenna array consists of two or more sparse subarrays featuring enhanced degrees of freedom (DOF) and reduced mutual coupling. This paper introduces a new class of planar coprime arrays, based on the theory of ideal lattices. In quadratic number fields, a splitting prime p can be decomposed into the product of two distinct prime ideals, which give rise to the two sparse subarrays. Their virtual difference coarray enjoys a quadratic gain in DOF, thanks to the generalized Chinese remainder theorem (CRT). To enlarge the contiguous aperture of the coarray, we present hole-free symmetric CRT arrays with simple closed-form expressions. The ring of Gaussian integers and the ring of Eisenstein integers are considered as examples to demonstrate the procedure of designing coprime arrays. With Eisenstein integers, our design yields a difference coarray that is a subset of the hexagonal lattice, offering a significant gain in DOF over the rectangular lattice, given the same physical areas. Maximization of CRT arrays in the aspect of essentialness and the superior performance in the context of angle estimation will be presented in the companion paper (Part II).

Reduced zero-divisor graphs of posets

This paper investigates properties of the reduced zero-divisor graph of a poset. We show that a vertex is an annihilator prime ideal if and only if it is adjacent to all other annihilator prime ideals and there are always two annihilator prime ideals which are not adjacent to a non-annihilator prime ideal. We also classify all posets whose reduced zero-divisor graph is planar or toroidal and the number of distinct annihilator prime ideals is four or seven.

Serial factorizations of right ideals

In a Dedekind domain D, every non-zero proper ideal A factors as a product A=P1t1⋯Pktk of powers of distinct prime ideals Pi. For a Dedekind domain D, the D-modules D/Piti are uniserial. We extend this property studying suitable factorizations A=A1…An of a right ideal A of an arbitrary ring R as a product of proper right ideals A1,…,An with all the modules R/Ai uniserial modules. When such factorizations exist, they are unique up to the order of the factors. Serial factorizations turn out to have connections with the theory of h-local Prüfer domains and that of semirigid commutative GCD domains.

The multiplicative ideal theory of Leavitt path algebras

It is shown that every Leavitt path algebra L of an arbitrary directed graph E over a field K is an arithmetical ring, that is, the distributive law A∩(B+C)=(A∩B)+(A∩C) holds for any three two-sided ideals of L. It is also shown that L is a multiplication ring, that is, given any two ideals A,B in L with A⊆B, there is always an ideal C such that A=BC, an indication of a possible rich multiplicative ideal theory for L. Existence and uniqueness of factorization of the ideals of L as products of special types of ideals such as prime, irreducible or primary ideals is investigated. The irreducible ideals of L turn out to be precisely the primary ideals of L. It is shown that an ideal I of L is a product of finitely many prime ideals if and only the graded part gr(I) of I is a product of prime ideals and that I/gr(I) is finitely generated with a generating set of cardinality no more than the number of distinct prime ideals in the prime factorization of gr(I). As an application, it is shown that if E is a finite graph, then every ideal of L is a product of prime ideals. The same conclusion holds if L is two-sided artinian or two-sided noetherian. Examples are constructed verifying whether some of the well-known theorems in the ideal theory of commutative rings such as the Cohen's theorem on prime ideals and the characterizing theorem on ZPI rings hold for Leavitt path algebras.

The Eisenstein ideal and Jacquet-Langlands isogeny over function fields

Let \mathfrak p and \mathfrak q be two distinct prime ideals of \mathbb{F}_q[T] . We use the Eisenstein ideal of the Hecke algebra of the Drinfeld modular curve X_0(\mathfrak p\mathfrak q) to compare the rational torsion subgroup of the Jacobian J_0(\mathfrak p\mathfrak q) with its subgroup generated by the cuspidal divisors, and to produce explicit examples of Jacquet-Langlands isogenies. Our results are stronger than what is currently known about the analogues of these problems over \mathbb Q .

Prime splitting in abelian number fields and linear combinations of Dirichlet characters

Let 𝕏 be a finite group of primitive Dirichlet characters. Let ξ = ∑χ∈𝕏 aχ χ be a nonzero element of the group ring ℤ[𝕏]. We investigate the smallest prime q that is coprime to the conductor of each χ ∈ 𝕏 and that satisfies ∑χ∈𝕏 aχ χ(q) ≠ 0. Our main result is a nontrivial upper bound on q valid for certain special forms ξ. From this, we deduce upper bounds on the smallest unramified prime with a given splitting type in an abelian number field. For example, let K/ℚ be an abelian number field of degree n and conductor f. Let g be a proper divisor of n. If there is any unramified rational prime q that splits into g distinct prime ideals in ØK, then the least such q satisfies [Formula: see text].

Prime Decompositions in Infinite Extensions of Global Fields

Let F be a global field with ring of integers A, and let K ⊆ L be a finite-dimensional Galois extension of fields algebraic over F, with B (resp., C) the integral closure of A in K (resp., L). If a nonzero prime ideal P of B has only one prime ideal 𝔓 of C lying over it and char(B/P) does not divide [L: K], then Gal(L/K) is a metacyclic group. Thus, if 0 < char(F) does not divide [L: K] and Gal(L/K) has a minimal generating set of cardinality at least 3, then each nonzero prime ideal P of B has at least two distinct prime ideals of C that lie over it. For any odd prime number p, an example is given of the latter situation where char(F) = p.

Multiplication modules with Krull dimension

In ring theory, it is shown that a commutative ring R with Krull dimension has classical Krull dimension and satisfies k.dim(R)=cl.k.dim(R). Moreover, R has only a finite number of distinct minimal prime ideals and some finite product of the minimal primes is zero (see Gordon and Robson [9, Theorem 8.12, Corollary 8.14, and Proposition 7.3]). In this paper, we give a generalization of these facts for multiplication modules over commutative rings. Actually, among other results, we prove that if M is a multiplication R-module with Krull dimension, then: (i) M is finitely generated, (ii) R has finitely many minimal prime ideals P_1, ..., P_n of Ann(M) such that P_1^k...P_n^kM=(0) for some k \geq 1, and (iii) M has classical Krull dimension and k.dim(M)=cl.k.dim(M)=k.dim(M/PM)= cl.k.dim(M/PM) for some prime ideal P of R.

ON A THEOREM OF DEDEKIND

Let K = ℚ(θ) be an algebraic number field with θ in the ring AK of algebraic integers of K and f(x) be the minimal polynomial of θ over the field ℚ of rational numbers. For a rational prime p, let [Formula: see text] be the factorization of the polynomial [Formula: see text] obtained by replacing each coefficient of f(x) modulo p into product of powers of distinct monic irreducible polynomials over ℤ/pℤ. Dedekind proved that if p does not divide [AK : ℤ[θ]], then the factorization of pAK as a product of powers of distinct prime ideals is given by [Formula: see text], with 𝔭i = pAK + gi(θ)AK, and residual degree [Formula: see text]. In this paper, we prove that if the factorization of a rational prime p in AK satisfies the above-mentioned three properties, then p does not divide [AK:ℤ[θ]]. Indeed the analogue of the converse is proved for general Dedekind domains. The method of proof leads to a generalization of one more result of Dedekind which characterizes all rational primes p dividing the index of K.

On 2-absorbing ideals of commutative rings

Suppose that R is a commutative ring with 1 ≠ 0. In this paper, we introduce the concept of 2-absorbing ideal which is a generalisation of prime ideal. A nonzero proper ideal I of R is called a 2-absorbing ideal of R if whenever a, b, c ∈ R and abc ∈ I, then ab ∈ I or ac ∈ I or bc ∈ I. It is shown that a nonzero proper ideal I of R is a 2-absorbing ideal if and only if whenever I1I2I3 ⊆ I for some ideals I1,I2,I3 of R, then I1I2 ⊆ I or I2I3 ⊆ I or I1I3 ⊆ I. It is shown that if I is a 2-absorbing ideal of R, then either Rad(I) is a prime ideal of R or Rad(I) = P1 ⋂ P2 where P1,P2 are the only distinct prime ideals of R that are minimal over I. Rings with the property that every nonzero proper ideal is a 2-absorbing ideal are characterised. All 2-absorbing ideals of valuation domains and Prüfer domains are completely described. It is shown that a Noetherian domain R is a Dedekind domain if and only if a 2-absorbing ideal of R is either a maximal ideal of R or M2 for some maximal ideal M of R or M1M2 where M1,M2 are some maximal ideals of R. If RM is Noetherian for each maximal ideal M of R, then it is shown that an integral domain R is an almost Dedekind domain if and only if a 2-absorbing ideal of R is either a maximal ideal of R or M2 for some maximal ideal M of R or M1M2 where M1,M2 are some maximal ideals of R.

A Generalization of Dedekind Criterion

Let K = ℚ(θ) be an algebraic number field with θ in the ring A K of algebraic integers of K and f(x) be the minimal polynomial of θ over the field ℚ of rational numbers. For a rational prime p, let (x) = (x) e 1 …(x) e r be the factorization of the polynomial (x) obtained by replacing each coefficient of f(x) modulo p into product of powers of distinct irreducible polynomials over ℤ/pℤ with g i (x) monic. In 1878, Dedekind proved that if p does not divide [A K :ℤ[θ]], then , where ℘1,…, ℘ r are distinct prime ideals of A K , ℘ i = pA K + g i (θ)A K with residual degree f(℘ i /p) = deg (x). He also gave a criterion which says that p does not divide [A K :ℤ[θ]] if and only if for each i, we have either e i = 1 or (x) does not divide where . The analog of the above result regarding the factorization in A K′ of any prime ideal 𝔭 of A K is in fact known for relative extensions K′/K of algebraic number fields with the condition “p ≠ | [A K :ℤ[θ]]” replaced by the assumption “every element of A K′ is congruent modulo 𝔭 to an element of A K [θ](†)”. In this article, our aim is to give a criterion like the one given by Dedekind which provides a necessary and sufficient condition for assumption (†) to be satisfied.

CONSTRUCTING CHAINS OF EXCELLENT RINGS WITH LOCAL GENERIC FORMAL FIBERS

ABSTRACT Let () be a complete local ring such that no integer is a zero divisor, , and the cardinality of the residue field is at least the cardinality of the real numbers. Let be a chain of distinct prime ideals of T such that intersected with the prime subring of T is the zero ideal and if Q is an associated prime ideal of T then Q is contained in . Then there exists a chain of local domains such that for every , the completion of is T and the generic formal fiber ring of is local with maximal ideal , where is the quotient field of . Furthermore, if, in addition to the above stated hypotheses, () is a complete regular local ring containing the rationals, then we can construct the chain so that each is excellent.

REPRESENTATIONS OF FINITE GROUPS AND HILBERT MODULAR FORMS

ABSTRACT Hecke studied the representation of in the space of elliptic modular cusp forms for the principal congruence subgroup with level odd prime . He showed that the difference of the multiplicities of certain two irreducible representations in this representation is the class number of . McQuillan extended Hecke's result to the case where the level of the principal congruence subgroup is , where are distinct odd primes. In the case of Hilbert modular forms for real quadratic fields, Meyer and Sczech gave a result which is analogous to that of Hecke. In this paper, we generalize their result to the case where the level of the principal congruence subgroup is , where are distinct prime ideals lying over odd primes. We get this result with the use of the holomorphic Lefschetz formula. Our result can be thought of a generalization of that of McQuillan to the real quadratic field case. We also give a generalization of a result of Eichler [3].

Constructing Local Generic Formal Fibers

Let (T,M) be a complete local (Noetherian) unique factorization domain of dimension at least two such that the cardinality of the residue fieldT/Mis at least the cardinality of the real numbers. Suppose (0)=p0⊂p1⊂···⊂pn≠Mis a chain of distinct prime ideals ofTsuch thatpnintersected with the prime subring ofTis the zero ideal. We construct a chain of local unique factorization domainsAn⊂An−1⊂···⊂A1⊂A0=Tsuch that the completion of eachAiisTand for everyithe generic formal fiber ofAiis local (this means that the ringT⊗AiKiis a local ring whereKiis the quotient field ofAi) withpi⊗AiKiits maximal ideal.

A characterization of finite Stone pseudocomplemented ordered sets

A distributive pseudocomplemented set $S$ [2] is called Stone if for all $a\in S$ the condition $LU(a^*,a^{**})=S$ holds. It is shown that in a finite case $S$ is Stone iff the join of all distinct minimal prime ideals of $S$ is equal to $S$.