Purity and Basic Subgroups

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).

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

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:

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.

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.

The construction of torsion-free abelian groups with prescribed endomorphism rings starting with Corner’s seminal work (see Corner, 1963) is a well-studied subject in the theory of abelian groups. Usually these construction work by adding elements from a (topological) completion in order to get rid of (kill) unwanted homomorphisms. The critical part is to actually prove that every unwanted homomorphism can be killed by adding a suitable element. We will demonstrate that some of those constructions can be significantly simplified by choosing the elements at random. As a result, the endomorphism ring will be almost surely prescribed, i.e., with probability one.

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.

Nowadays purity plays a role in at least four branches of mathematics: Module Theory, Theory of Acts over semigroups, Model Theory, and Category Theory. Adamek and Rosciký have studied these notions categorically, and Rothmaler model-theoretically. Some authors including Banaschewski, Gould, and Normak have studied purity on G-acts, acts over a monoid or a group G.

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.

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.

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).

CHAPTER V - BASIC SUBGROUPS

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