Rigid abelian groups and the probabilistic method

On the computational complexity of the theory of Abelian groups

We shall characterize the complete first order theories of torsion free abelian groups having algebraically prime models, elementarily prime models, minimal models, etc., using the well known invariants from [9], [3]. This study was suggested by the questions raised in [2]. Group here will always mean abelian group. For reference on groups see Fuchs [5], [6], for model theory Chang-Keisler [1]. All theories considered are theories in the first order language of group theory. A group G is minimal (strictly minimal) if H < G implies H = G (H-= G), and elementarily prime (algebraically prime) if G is elementarily (algebraically) embeddable in every H G. p, q always denote prime numbers, P the set of all primes. The p-rank of a group G, p-rk(G), is the dimension of G/pG as a vector space over Z/pZ. We define tf(p, G): = min {p-rk(G), ~o}. By the results of [9], [3], two nontrivial torsion free abelian groups G and H are elementarily equivalent iff for all primes p tf(p, G) = tf(p, H). This gives a natural 1-1 correspondence between the complete theories of nontrivial torsion free abelian groups and the functions /3 : P --~ ~o + 1. /3 will always stand for such a function, and the corresponding complete theory will be denoted by T 0. S o is the set of primes p for which/3(p) ~ 0. For more definitions see section 3. Let 13 be a function from P to o) + 1. We shall study the following questions: Does T o have an elementarily (algebraically) prime model? Does T o have a minimal (strictly minimal) model, and if this is the case, how many such models does T o have? Do the minimal and strictly minimal models of T o coincide? These questions have been studied for the special case of the theory of the

It was shown in Huber [4; Theorem 3.3] that an R-module E is slender as a module over its endomorphism ring S = EndR E if it is of the form E = M(IN) for some R-module M. This had important consequences for questions of E-reflexivity. In Mader [6; Lemma 5.1] it was shown that the abelian group E = M ∞i=1 Z(pi)is slender over S = End E and this was a key step in proving that the completion of a group A of non-measurable cardinality equipped with the direct-sum-of-cyclics topology of D’Este [1] is algebraically the group HomS(HomZ (A, E), E). At the Honolulu Conference on Abelian Group Theory 1982/83, Martin Huber suggested to me that [6; Lemma 5.1] should be extendable to more general direct sums. In this note we prove such a theorem and use it to characterize completely the totally projective p-groups which are slender as modules over their endomorphism rings.

In the theory of abelian groups, the study of relations between an abelian group and its endomorphism group is an important problem. The question arises on conditions under which an abelian group and its endomorphism group are isomorphic ([1], Problem 31). The answer to this question has been obtained in the classes of completely decomposable abelian groups [2], vector groups [3], in the classes of periodic abelian groups, divisible abelian groups, nonreduced abelian groups, reduced algebraically compact abelian groups [4]. A group A is called an E+-group if it is isomorphic to its endomorphism group End(A). Clearly, if an abelian group A is an E+-group, then its endomorphism group End(A) is also an E+-group. In this paper, we show that the converse statement holds not always. A group A is called an End-group if the following implication takes place:

The design of decision procedures for combinations of theories sharing some arithmetic fragment is a challenging problem in verification. One possible solution is to apply a combination method à la Nelson-Oppen, like the one developed by Ghilardi for unions of non-disjoint theories. We show how to apply this non-disjoint combination method with the theory of abelian groups as shared theory. We consider the completeness and the effectiveness of this non-disjoint combination method. For the completeness, we show that the theory of abelian groups can be embedded into a theory admitting quantifier elimination. For achieving effectiveness, we rely on a superposition calculus modulo abelian groups that is shown complete for theories of practical interest in verification.

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.

In 1996, the first author defined analogs of the concepts of complete (divisible), reduced, and periodic abelian groups, well-known in the theory of abelian groups, for arbitrary varieties of algebras. In 2021, the first author proposed a modification of the concepts of completeness and reducibility, which is more natural in the case of associative rings. The paper studies the modification of these concepts for associative rings. Artinian \(\mathbf{M}\)-complete, \(\mathbf{M}\)-reduced rings, and minimally \(\mathbf{M}\)-complete associative nilpotent rings, simple rings with unity, and finite rings are characterized.

It is well known that assuming Martin’s axiom and the negation of the continuum hypothesis (MA+−1CH) has a dramatic effect on Abelian group theory. One only has to think of Shelah’s resolution of the Whitehead problem (cf[S1] or [E1]). MA+−NCH has certain drawbacks. It is not strong enough to resolve questions about the structure of ω1-separable groups which can be resolved by an extension and it is not consistent with the continuum hypothesis. In this paper I shall talk about generalizations due to Shelah of MA+−ICH, the proper forcing axioms, and their relation to Abelian group theory. (From hereon “group” will mean “Abelian group”.) Section I is devoted to some preliminaries and the statement of the proper forcing axioms. In section II a useful result about choosing a cub almost disjoint from a ladder system assuming PFA(ω1) is proved in order to illustrate how the axiom works. Proper posets are redefined in section III using elementary submodels. This section simplifies the task of writing the remainder of the paper. But it is explained for the reader who is unversed in logic how to ignore this section.KeywordsDirect SummandDense SubsetContinuum HypothesisPure SubgroupMaximal AntichainThese 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.

Assuming the Axiom of Constructibility, necessary and sufficient conditions are given for the vanishing of Ext\ for rings A of global dimension 1.Using Martin's Axiom, the necessity of these conditions is shown not to be a theorem of ZFC.Applications are given to abelian group theory, including a partial solution (assuming V = L) to a problem of Baer on the splitting of abelian groups.Some independence results in abelian group theory are also proved.Introduction.In a remarkable recent paper [22], S. Shelah proved that Whitehead's problem is unsolvable in ordinary set theory (Zermelo-Frankel set theory with the Axiom of Choice).More precisely, he proved that two different answers to the problem are obtained when set theory is extended by adding two different axioms: the Axiom of Constructibility and Martin's Axiom.(See Theorem 4.1.)Shelah's arguments were specific to Whitehead's problem and nonhomological in character.In this paper we extend Shelah's techniques in order to obtain some general results about the vanishing of ExtA (for modules over a ring A of global dimension 1) in different models of set theory, and we apply them to some specific problems in abelian group theory.Assuming the Axiom of Constructibility we obtain necessary and sufficient conditions for the vanishing of ExtA (Theorems 1.2 and 1.5).To state the result in.a special case: if C and A are abelian groups of cardinality N,, Ext(C, A) -0 if and only if C is the union of an increasing chain of countable subgroups [Cv\v < «,} such that the chain is smooth (i.e., C0 -U"<oC, for aii iimit ordinals a) and satisfies Ext(C0, A) = 0 and E\i(Cv+x/Cv,A) = 0 for all v < K,.Although the sufficiency of this condition is an easy consequence of classical homological algebra (see Theorem 1.2), the proof of necessity makes use of the Axiom of Constructibility (Theorem 1.5).The necessity of this condition is not provable in ordinary set theory; in particular the condition is not necessary when Martin's Axiom is assumed ( §3).

Stable theories were introduced in [1] for constructing classification theory and are a generalization of the concept of a totally transcendental theory as defined in [2]. In [3], it was proved that the property of being stable for a theory is equivalent to being definable for every complete type of the theory. This property plays a fundamental part in research on stable theories. In [4], the notion of E∗-stability (generalized stability) was introduced, and it was proved that types for E∗-stable theories are definable. A consequence of that result was stating, along with definability of types for stable theories (see [3]), that types over any P -sets in P -stable theories likewise are definable (which had been established in [5] for types over P -models). The notion of E∗-stability is a new stability scale, whose basic parameter is a mapping of types of a complete theory into types of another theory. An interesting example of E∗-stability is (P, a)-stability, defined by adding to a language a unary predicate symbol and adding to types the condition of being algebraically closed for that predicate. In this paper, we work to prove a (P, a)-stability theorem for theories of torsion-free Abelian groups. In so doing, use is made of quantifier elimination down to positive primitive formulas in Abelian groups with a predicate distinguishing a subgroup. By virtue of this fact, the question of being (P, a)-stable for a theory reduces to asking if a system of linear equations with integer coefficients has a solution in the algebraic closure of constants involved in the system. To ∗Supported by RFBR (project No. 09-01-00336-a).

(1984). The theory of separable mixed abelian groups. Communications in Algebra: Vol. 12, No. 15, pp. 1813-1834.

In this chapter, we are going to discuss a basic concept: pure subgroup. This concept has been one of the most fertile notions in the theory since its inception in a paper by the pioneer H. Prufer. The relevance of purity in abelian group theory, and later in module theory, has tremendously grown with time. While abelian groups have been major motivation for a number of theorems in category theory, purity has served as a prototype for relative homological algebra, and has played a significant role in model theory as well.

Let R be an associative ring with a unit and N be a left R-module. The set MR(N) = {f: N → N | f(rx) = rf(x), r ∈ R, x ∈ N} is a near-ring with respect to the operations of addition and composition and contains the ring ER(N) of all endomorphisms of the R-module N. The R-module N is endomorphic if MR(N) = ER(N). We call an Abelian group endomorphic if it is an endomorphic module over its endomorphism ring. In this paper, we find endomorphic Abelian groups in the classes of all separable torsion-free groups, torsion groups, almost completely decomposable torsion-free groups, and indecomposable torsion-free groups of rank 2.

In this article we prove a theorem on the definitional embeddability into first-order predicate logic without equality of such well-known mathematical theories as group theory and the theory of Abelian groups. This result may seem surprising, since it is generally believed that these theories have a non-logical content. It turns out that the central theory of general algebra are purely logical. Could this be the reason that we find them in many branches of mathematics? This result will be of interest not only for logicians and mathematicians but also for philosophers who study foundations of logic and its relation to mathematics.

The projective dimension of an abelian group with torsion reduced part, as a module over its endomorphism ring, is determined.In particular, a group of projective dimension 2 is exhibited.Any abelian group G may be considered as a module over its endomorphism ring E. It then is natural to study the properties of G as an P-module.In particular, you can try to compute the various homological dimensions of G.For example, in [6] we showed that a /J-group G is flat over E (i.e. has weak dimension zero) unless its reduced part is bounded and its divisible part is nonzero.In [7] we characterized those G which are injective over P. Douglas and Farahat [2] showed that if G is torsion, then dimEG < 1, where dim£G denotes the projective dimension of G over P. Later [3] they