Groups and Modules that are Slender as Modules over Their Endomorphism Rings

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.

On the computational complexity of the theory of Abelian groups

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 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

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.

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.

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

If E is a ring, and G is an E-module, then G is quasi-injective if every homomorphism from a submodule of G to G can be extended to an endomorphism of G. POOLE and REID [2] raise the question as to which abelian groups G are quasiinjective as modules over their endomorphism rings E. They prove that direct sums of cyclic p-groups are quasi-injective. This result is extended here to a large class of p-groups, including all p-groups that have no elements of infinite height, and all totally projective p-groups -- in particular, all countable p-groups. The central idea is a property shared by totally projective p-groups and pgroups with no elements of infinite height. DEFINITION. If G is a p-group and xEG[p], then x is said to be detachable if G can be written as G1 @ Gz with (x) =p'G~ [p] for some ordinal c~. If every element of G[p] is detachable, we say that G is detachable. We write p'G~[p] instead of p~G~ to allow Ga to be divisible. Note that every element in G[p] of finite height is detachable, so every p-group with no elements of infinite height is detachable. Hill's extension of Ulm's theorem to totally projective p-groups (see [3]) implies that these groups are also detachable. THEOREM 1. If G is a detachable p-group, and x, yEG[p], then there is an endomorphism f of G such that f(x) =y or f(y) = x.

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.

Properties of abelian groups related to a given finite rank torsion free abelian group A are analyzed in terms of End (A), the endomorphism ring of A. This point of view gives rise to generalizations of some classical theorems by R. Baer and examples of pathological direct sum decompositions of finite rank torsion free abelian groups.

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).

We prove in this note that the ring of endomorphisms of an infinite bounded Abelian group admits a nondiscrete right bounded ring topology. We give an example of an Abelian group whose ring of endomorphisms admits both nondiscrete left and right bounded topologies but does not admit a nondiscrete bounded ring topology.

In a recent paper2 we have called a ring 2 primitive if 2 contains a maximal right ideal a such that the quotient (a: W) = (0). The quotient (: 1) is defined to be the largest two-sided ideal of 21 that satisfies the condition 21(a: 21) C a. Thus if 21lias an identity, (a: 21) is the largest two-sided ideal of 21 contained in a. Primitive rings appear to play a role in the general structure theory of rings analogous to that played by simple rings in the Wedderburn-Artin theory. Primitive rings are also fundamental in general representation theory since it is known that a necessary and sufficient condition that a ring 21 be primitive is that 21 be isomorphic to an irreducible ring W of endomorphisms in a commutative group T. If 2 is irreducible in T and Zi is the division ring of endomorphisms commutative with the elements of A, then R is a dense ring of linear transformations in T regarded as a vector space over Z.3 We shall call 2 a left primitive ring if 2 contains a maximal left ideal a' such that the quotient (}': 21)l = (0). Here (s': 2) 1 is the largest two-sided ideal in 2 having the property (a': 21)1 21 C a'. Evidently, a ring is left primitive if and only if it is anti-isomorphic to an irreducible ring of endomorphisms. It is an open question whether or not a primitive ring is necessarily left primitive. This is indeed the case if 2 contains minimal ideals. In many other respects the theory of primitive rings that contain minimal ideals constitutes the most satisfactory part of the theory of primitive rings. An appropriate tool for studying rings of this type is a generalization of the duality theory previously used by Dieudonn6 in studying simple rings that possess minimal ideals.4 In this note we develop this generalization of Dieudonn6's results. We also investigate certain natural topologies that can be defined in any primitive ring.