On Dedekind domains whose class groups are direct sums of cyclic groups

All groups herein are assumed to be abelian. It was not until the 1940's that it was known that a subgroup of an infinite direct sum of finite cyclic groups is again a direct sum of cyclics. This result rests on a general criterion due to Kulikov [7] for a primary abelian group to be a direct sum of cyclic groups. If G is p-primary, Kulikov's criterion presupposes that G has no elements (other than zero) having infinite p-height. For such a group G, the criterion is simply that G be the union of an ascending sequence of subgroups Hn where the heights of the elements of Hn computed in G are bounded by some positive integer λ(n). The theory of abelian groups has now developed to the point that totally projective groups currently play much the same role, at least in the theory of torsion groups, that direct sums of cyclic groups and countable groups played in combination prior to the discovery of totally projective groups and their structure beginning with a paper by R. Nunke [11] in 1967.

The study of factorization in integral domains has played an important role in commutative algebra for many years. Much of this work has con­centrated on the study of unique factorization domains (UFDs). Beyond the realm of UFDs, there is a large class of integral domains for which each nonunit can be factored as a product of irreducibles, yet the factorization may not be unique. Classically, Dedekind domains are such domains. A Dedekind domain D is a UFD if and only if its ideal class group is trivial. Hence the size of the class group becomes a measure of how far the domain D is from having “unique factorization”. The concept of the class group and its relation to factorization properties extends also to the more general class of Krull domains. It is within the context of these domains that we consider in this paper general questions concerning the lengths of factorization in the non-UFD setting.

The study of effectiveness in classical mathematics is rapidly expanding, through recent research in algebra, topology, model theory, and functional analysis. Well-known contributors are Barwise (Wisconsin), Crossley (Monash), Dekker (Rutgers), Ershoff (Novosibirsk), Feferman (Stanford), Harrington (Berkeley), Mal′cev (Novosibirsk), Morley (Cornell), Nerode (Cornell), Rabin (Hebrew University), Shore (Cornell). Further interesting work is due to Kalantari (University of California, Santa Barbara), Metakides (Rochester), Millar (Wisconsin), Remmel (University of California, San Diego), Nurtazin (Novosibirsk). Areas investigated include enumerated algebras, models of complete theories, vector spaces, fields, orderings, Hilbert spaces, and boolean algebras.We investigate the effective content of the structure theory of p-groups. Recall that a p-group is a torsion abelian group in which the (finite) order of each element is some power of a fixed prime p. (In the sequel, “group” = “additively written abelian group”.)The structure theory of p-groups is based on the two elementary notions of order and height. Recall that the order of x is the least integer n such that nx = 0. The height of x is the number of times p divides x, that is, the least n such that x = pny for some y in the group but x ≠ pn+1y for any y. If for each n ∈ ω there is a “pnth-root” yn, so that x = pnyn, then we say that x has infinite height. In 1923, Prüfer related the two notions as criteria for direct sum decomposition, provingTheorem. Every group of bounded order is a direct sum of cyclic groups, andTheorem. Every countable primary group with no (nonzero) elements of infinite height is a direct sum of cyclic groups.

The study of Hopficity in Abelian groups has been largely motivated by the fundamental results of Baumslag, who proved that torsion groups are always Hopfian regardless of their cardinality, but left several questions open concerning torsion-free groups. Later, Corner addressed some of these questions by providing counterexamples showing that a direct sum of two Hopfian groups can be non-Hopfian, and that a group with an automorphism group of order two does not guarantee Hopficity. These results highlighted the need for new constructions to explore Hopficity in torsion-free Abelian groups. Our work introduces a new approach based on divisibility techniques, as it contributes to the understanding of free-torsion groups with respect to the Hopficity property, providing new insights into their structural properties and implications within group theory. Our analysis also demonstrates how divisibility properties, as well as the introduction of totally invariant subgroups and homomorphisms, can be used to establish Hopficity in specific Abelian groups, particularly those that are free-torsion. In order to reach all of this, we start by taking a group defined as an infinite direct sum of cyclic groups; then we construct a specific subgroup generated by two particular families of elements; and finally we show that this group is Hopfian through results from the theory of divisible subgroups.

The fundamental theorem of abelian groups states that any finitely generated abelian group is a direct sum of cyclic groups. This theorem plays a fundamental role in the structure theory of abelian groups. It has fascinated many algebraists to look at this theorem from different points of view:- (a) To find suitable generalizations of this theorem for modules over certain classes of rings, e.g. Dedekind domain, hereditary noetherian prime rings, valuation rings etc. (b) To create suitable versions of this theorem in some module categories and use such versions to develop the structure theory of such module categories. For example, torsion abelian group-like modules, modules with finitely generated submodules direct sums of multiplication modules. (c) To find those rings for which certain versions of the fundamental theorem of abelian groups hold. For example rings over which all finitely generated modules are direct sums of cyclic modules. (d) To examine the structure of certain classes of abelian groups and to try to find modules with similar structures. (e) To find roles of answers to some of above types of questions in the general theory of modules.

This chapter develops tools for a more penetrating study of algebraic number theory than was possible in Chapter V and concludes by formulating two of the main three theorems of Chapter V in the modern setting of “adeles” and “ideles” commonly used in the subject.Sections 1–5 introduce discrete valuations, absolute values, and completions for fields, always paying attention to implications for number fields and for certain kinds of function fields. Section 1 contains a prototype for all these notions in the construction of the fieldQp of p-adic numbers formed out of the rationals. Discrete valuations in Section 2 are a generalization of the order-of-vanishing function about a point in the theory of one complex variable. Absolute values in Section 3 are real-valued multiplicative functions that give a metric on a field, and the pair consisting of a field and an absolute value is called a valued field. Inequivalent absolute values have a certain independence property that is captured by the Weak Approximation Theorem. Completions in Section 4 are functions mapping valued fields into their metric-space completions. Section 5 concerns Hensel’s Lemma, which in its simplest form allows one to lift roots of polynomials over finite prime fields Fp to roots of corresponding polynomials over p-adic fields Qp.Section 6 contains the main theorem for investigating the fundamental question of how prime ideals split in extensions. Let K be a finite separable extension of a field F, let R be a Dedekind domain with field of fractions F, and let T be the integral closure of R in K. The question concerns the factorization of an ideal pT in T when p is a nonzero prime ideal in R. If Fp denotes the completion of F with respect to p, the theorem explains how the tensor product K ⊗F Fp splits uniquely as a direct sum of completions of valued fields. The theorem in effect reduces the question of the splitting of pT in T to the splitting of Fp in a complete field in which only one of the prime factors of pT plays a role.Section 7 is a brief aside mentioning additional conclusions one can draw when the extension K/F is a Galois extension.Section 8 applies the main theorem of Section 6 to an analysis of the different of K/F and ultimately to the absolute discriminant of a number field. With the new sharp tools developed in the present chapter, including a Strong Approximation Theorem that is proved in Section 8, a complete proof is given for the Dedekind Discriminant Theorem; only a partial proof had been accessible in Chapter V.Sections 9–10 specialize to the case of number fields and to function fields that are finite separable extensions of Fq (X), where Fq is a finite field. The adele ring and the idele group are introduced for each of these kinds of fields, and it is shown how the original field embeds discretely in the adeles and how the multiplicative group embeds discretely in the ideles. The main theorems are compactness theorems about the quotient of the adeles by the embedded field and about the quotient of the normalized ideles by the embedded multiplicative group. Proofs are given only for number fields. In the first case the compactness encodes the Strong Approximation Theorem of Section 8 and the Artin product formula of Section 9. In the second case the compactness encodes both the finiteness of the class number and the Dirichlet Unit Theorem.KeywordsPrime IdealValuation RingBasic AlgebraDedekind DomainFractional IdealThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Introduction.Given a field k and a fc-algebra A, Gerstenhaber has introduced the concept of a deformation of A [8] which is a [[/]]-algebra structure on the module A [[/]].One of the main problems of the theory of deformations is to determine those algebras for which the only deformation (modulo an equivalence relation) is the usual A-[[r]]-algebra structure on .4[[*]].Such algebras are said to be rigid.If A is a commutative /c-algebra, we may restrict our attention to those deformations of A which are commutative.This leads us to consider the commutative cohomology groups of A, Hn(A, M)s, defined by Harrison [11].The principal purpose of this paper is to investigate the following conditions on a commutative algebra A over a field k:(1) H2(A, M)s = 0 for all ^-modules M.(2) H\A,f=0.(3) A is a rigid /r-algebra in the commutative deformation theory.(4) The module of differentials of A over k is a projective /4-module.(5) A is a finite product of (not necessarily algebraic) separable extension fields of k.We may show that (1) implies (4) [9, Theorem (20.4.9)] and that (2) implies (3) [8, p. 65].Condition (1) may be expressed by saying that A is a formally smooth fc-algebra [9, 19.4.4].When A is a field, Gerstenhaber has shown that (1), (2), and ( 5) are equivalent and he conjectured that these conditions were equivalent to (3).We show that (1), ( 2), (3), and (5) are equivalent even for A semisimple (Theorem 6.5).Moreover, if A is artinian, then (1) and ( 5) are equivalent (Proposition 5.9).In another direction, if A is a geometric complete intersection, then (1), (2), and (3) are equivalent (Theorem 6.8a).If in addition A is an integral domain with a separable quotient field, then (1), ( 2), (3), and (4) are equivalent (Theorem 6.8c).In the final section, we investigate the above conditions for the group algebra B[G] of an abelian group G which is a direct sum of cyclic groups with B a commutative algebra over a field k of positive characteristic p.In addition, we consider the following condition on G and k:(6) G has no element of order p = char (k).

an almost Dedekind domain; one of these conditions is that l) primary ideals are Dedekind ideals, and 2) proper prime ideals in D are maximal

Suppose that G is a p-primary abelian group. The subgroup G[p] = {x∈G:px=0} is called the socle of G and any subgroup S of G[p] is called a subsocle of G. If each subsocle of G supports a pure subgroup, then G is said to be pure-complete [1]. It is well known that, if G a direct sum of cyclic groups, then G is necessarily pure-complete. Further results about pure-complete groups are contained in [1] and [3].

CHAPTER V - BASIC SUBGROUPS

A cotorsion hull of the separable 𝑝-group 𝑇 is considered when 𝑇 is a direct sum of torsion-complete groups. It is proved that in the considered case its cotorsion hull is fully transitive if and only if 𝑇 is a direct sum of cyclic groups or is a torsion-complete group.

Necessary conditions are given in order that Tor(Λ, B) be a direct sum of cyclic groups for abelian groups A and B. Sufficient conditions are also given that compare with the necessary conditions.They are only slightly stronger if at all, and the two are equivalent for groups of cardinality not exceeding K 2 * Due to the complexity of the problem, these conditions are not absolute, but constitute a reduction to smaller cardinality.The results generalize earlier results of R. Nunke.What is known of the structure and properties of Tor(^4, B) for abelian groups A and B is primarily due to R. Nunke.The main results appear in [4], [5], and [6].Contributions of other authors, however, were manifested in [2] and [3].This note generalizes the results of Nunke published in [4] and [6] concerning a basic question: When is Tor(^4, B) a direct sum of cyclic groups?Nunke has provided a completely satisfactory answer when neither the cardinality of A nor B exceeds 8j and in certain other cases, as well, allowing A and B to be arbitrarily large.However, the general case where A or B has cardinality greater than # λ was deferred.In this paper, we settle the question for cardinality X 2 .For the general case, we give necessary conditions in order that Ύoτ{A, B) be a direct sum of cyclic groups, and \γe also give sufficient conditions that are very close to the necessary ones, but the gap is not bridged completely.Since Tor(^4, B) is not affected by the nontorsion portions of A and 5, we can assume without loss of generality that A and B are torsion.Further, we can specialize, as usual, to the case that A and B are both /7-primary.Therefore, all groups are assumed to be/^-primary.The following three theorems summarize the major known results concerning the question of when Tor(^4, B) is a direct sum of cyclic groups.Following [6], we say that G is Σ-cyclic if G is a direct sum of cyclic groups.THEOREM A {Nunke [4, Corollary 3.5]).If p ω A φ 0, then Tor(^, B) is Σ-cyclic only if B is Έ-cyclic.The preceding result also appears in [6] as Theorem 12(i).

A solid mathematical technique for handling uncertainty is offered by the Zadeh-conceived notion of fuzzy sets, provides a convenient point of departure for the construction of a conceptual framework. In this paper, we explore the advantages of applying techniques from fuzzy set theory, paving the way for innovative approaches in the study of fuzzy multiplicative ideal theory. This study investigates that every nonzero fuzzy fractional ideal in a Dedekind domain has a unique factorization into fuzzy prime ideals. We demonstrate a few intermediate findings about the localization of a noetherian domain at its fuzzy fractional ideals. Introducing the new concepts and provide some examples via discrete valuation ring illustrates that a discrete valuation ring is the localization of a noetherian domain at its fuzzy fractional ideals.

Assuming Generalized Riemann’s Hypothesis, Bach proved that the class group C ℓ K \mathcal C\!\ell _{\mathbf {K}} of a number field K \mathbf {K} may be generated using prime ideals whose norm is bounded by 12 log 2 ⁡ Δ K 12\log ^2\Delta _{\mathbf {K}} , and by ( 4 + o ( 1 ) ) log 2 ⁡ Δ K (4+o(1))\log ^2\Delta _{\mathbf {K}} asymptotically, where Δ K \Delta _{\mathbf {K}} is the absolute value of the discriminant of K \mathbf {K} . Under the same assumption, Belabas, Diaz y Diaz and Friedman showed a way to determine a set of prime ideals that generates C ℓ K \mathcal C\!\ell _{\mathbf {K}} and which performs better than Bach’s bound in computations, but which is asymptotically worse. In this paper we show that C ℓ K \mathcal C\!\ell _{\mathbf {K}} is generated by prime ideals whose norm is bounded by the minimum of 4.01 log 2 ⁡ Δ K 4.01\log ^2\Delta _{\mathbf {K}} , 4 ( 1 + ( 2 π e γ ) − N K ) 2 log 2 ⁡ Δ K 4\big (1+\big (2\pi e^{\gamma })^{-N_{\mathbf {K}}}\big )^2\log ^2\Delta _{\mathbf {K}} and 4 ( log ⁡ Δ K + log ⁡ log ⁡ Δ K − ( γ + log ⁡ 2 π ) N K + 1 + ( N K + 1 ) log ⁡ ( 7 log ⁡ Δ K ) log ⁡ Δ K ) 2 4\big (\log \Delta _{\mathbf {K}} +\log \log \Delta _{\mathbf {K}}-(\gamma +\log 2\pi )N_{\mathbf {K}}+1+(N_{\mathbf {K}}+1)\frac {\log (7\log \Delta _{\mathbf {K}})} {\log \Delta _{\mathbf {K}}}\big )^2 . Moreover, we prove explicit upper bounds for the size of the set determined by Belabas, Diaz y Diaz and Friedman’s algorithms, confirming that it has size ≍ ( log ⁡ Δ K log ⁡ log ⁡ Δ K ) 2 \asymp (\log \Delta _{\mathbf {K}}\log \log \Delta _{\mathbf {K}})^2 . In addition, we propose a different algorithm which produces a set of generators which satisfies the above mentioned bounds and in explicit computations turns out to be smaller than log 2 ⁡ Δ K \log ^2\Delta _{\mathbf {K}} except for 7 7 out of the 31292 31292 fields we tested.

Primary decomposition of the J-groups of complex projective and lens spaces