When Tor(A, B) is a direct sum 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 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.

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.

For a given family (Gi)i∈N of finitely generated abelian groups, we construct a Dedekind domain D having the following properties.(1)Pic(D)≅⨁i∈NGi.(2)For each i∈N, there exists a submonoid Si⊆D• with Pic(DSi)≅Gi.(3)Each class of Pic(D) and of all Pic(DSi) contains infinitely many prime ideals. Furthermore, we study orders as well as sets of lengths in the Dedekind domain D and in all its localizations DSi.

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.

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

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.

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).

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

We study nonlinear networks invariant under groups (mostly cyclic) of operations. We deduce the consequences of invariance on the reduced incidence matrix A, the branch admittance operator <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">{\cal Y}_{n}</tex> , and the node admittance operator <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">{\cal Y}_{n}</tex> . We consider two special kinds of excitation: symmetric (for any groups) and alternating (for cyclic groups). Under the uniqueness assumption, the solutions are shown to be symmetric and alternating respectively, and the equations required for solving the network are considerably simplified. An example shows that if uniqueness does not hold, a symmetric excitation can give rise to symmetric and nonsymmetric solutions! The case of linear networks is treated as a special case of the nonlinear networks and the decoupling property is easily obtained. For the nonlinear case, we show by examples that except for some special interconnections of nonlinear elements, the function space <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L^{n}</tex> cannot be decomposed into direct sum of subspaces which are invariant under the map <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">{\cal Y}_{n}</tex> ,. Finally we derive stability conditions for three kinds of periodic oscillations; symmetric oscillation, alternating oscillation, and oscillation with delay, where the last two cases are restricted to cyclic groups. It is shown how the stability conditions can be systematically simplified.

Having reduced the general theory of the FT on finite Abelian groups to the theory of the FT on finite cyclic groups, it suffices to study the FT of functions defined on the cyclic group Zn for an arbitrary value of n, where n > 1. Our next goals are the following: Determine the form of eigenvectors of the FT (Section 7.2). Find the spectral decomposition of the FT on Zn or, equivalently, the decomposition of the space VZn as a direct sum of its invariant subspaces (Section 7.3). Determine the multiplicity of the eigenvalues of the FT (Section 7.5).

On the computational complexity of the theory of Abelian groups

Definite results are obtained in the conditions: the group G is torsion free and the class ~h of the extension 0 ~ Zh ~ G h --* N h --+ 0 in H2(Nh, ) @ k is niipotent (here Gh, for h~ G, stands for the centralizer of h). The cyclic homology groups HC,(A >~ G) decompose naturally as a direct sum of two subgroups, called "the homogeneous" and "the inhomogeneous" parts ofHC, (A >~ G). The homogeneous part can be obtained from a sPectral sequence converging to it with E2,q = Hp(G, HCq(A)). The inhomogeneous part vanishes after inverting S. For G -- Z and in the topological situation we thus reobtain the result of Nest [17]. For general G but A = k and trivial action, thus for group rings k[G], we reobtain results of Burghelea [4]. The free groups satisfy our conditions. (See [10] for other classes of groups satisfying these conditions.) In this case our results are compatible with the results of Pimsner and Voiculescu [ 18; 19] and suggest that, for a large class of groups, the ~-part of KtOr'(A>~r~d G) is obtained from a convergent spectral sequence with E~.q = Hp(G, K~~ Here ..,/r176 denotes the topological K-theory functors (see [2], A is a C*-algebra and A >%,d G, the reduced crossed product C*-algebra, is a suitable completion of A >~G. Kasparov [13] succeeded to prove this for groups having a "special manifold" as classifying space. We expect that our spectral sequence will give more insight in Kasparov's spectral sequence. We also mention

Introduction. In [7] and [8] we have determined the isomorphisms of the endomorphism rings of two special classes of torsion-free modules. In [7] the modules were free, over principal left ideal domains, and the isomorpnism was always induced by a semi-linear transformation of the modules. In [8] the ring of scalars was a complete discrete valuation ring, but no further restriction was placed on the modules. Except for the special case when both modules are divisible, the same result holds as in [7]. In each case a key point in the argument is that a certain indecomposable direct summand can be shown to be cyclic. Even for ordinary torsion-free abelian groups, the abundance of the indecomposable ones makes it impossible to prove that isomorphism of the endomorphism rings implies isomorphism of the underlying groups. We show first that for any cardinal number c, there exist homogeneous completely decomposable groups G and H of rank c which are not isomorphic, but have isomorphic rings of endomorphisms. This makes it clear that strong restrictions must be placed on the class of groups considered if it is hoped that the ring isomorphism is to be induced by a group isomorphism. In particular, homogeneity coupled with separability or complete decomposability is not sufficient. We are, however, able to generalize the results of [7] to a class of what we call locally free modules A over principal left ideal domains F. A locally free abelian group (F is the ring of integers) is just a homogeneous separable group of null type [6, p. 208]. This latter class of groups includes the complete direct sum of infinite cyclic groups, as well as all its subgroups [4, p. 176]. For example, the set of all bounded sequences of integers is such a group. We also determine (Theorem B) which semi-linear transformations induce the same ring isomorphisms. The results are similar to the vector space case considered in [2], and apply to the endomorphism rings of locally free modules as well as the torsion-free modules of [8].