Whitehead modules

The Baer and the quasi-Baer properties of rings are extended to a module theoretic setting in this chapter. Using the endomorphism ring of a module, the notions of Baer, quasi-Baer, and Rickart modules are introduced and studied. Similar to the fact that every Baer ring is nonsingular, we shall see that every Baer module satisfies a weaker notion of nonsingularity of modules (\(\mathcal{K}\)-nonsingularity) which depends on the endomorphism ring of the module. Strong connections between a Baer module and an extending module will be observed via this weak nonsingularity and its dual notion. It is shown that an extending module which is \(\mathcal{K}\)-nonsingular is precisely a \(\mathcal{K}\)-cononsingular Baer module. This provides a module theoretic analogue of the Chatters-Khuri theorem for rings. Direct summands of Baer and quasi-Baer modules respectively inherit these properties. This provides a rich source of examples of Baer and quasi-Baer modules, since one can readily see that for any (quasi-)Baer ring R and an idempotent e in R, the right R-module eR R is always a (quasi-)Baer module. It will be seen that every projective module over a quasi-Baer ring is a quasi-Baer module. Connections of a (quasi-)Baer module and its endomorphism ring are discussed. Characterizations of classes of rings via the Baer property of certain classes of free modules over them are presented. An application also yields a type theory for \(\mathcal{K}\)-nonsingular extending (continuous) modules which, in particular, improve the type theory for nonsingular injective modules provided by Goodearl and Boyle.

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).

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.

For a commutative ring A we study here the Grothendieck group, K0A, of finitely generated projective A-modules, and the Picard group, Pic (A), of projective modules of rank one (under (?A). Writing koA for the elements of rank zero in KOA, there is an epimorphism det: K0A Pic (A) defined by the rth exterior power of a projective module of rank r. Our original objective was to calculate KJ(Zw), where w is a finitely generated abelian group. We do this in ? 8. The bulk of the paper is a rather general discussion of the techniques, some already familiar in other contexts, which we use to make this calculation. Our procedure is to decompose w as w0 x T, with w0 a finite group and T free abelian, and then to write Zw= A[T] where A = Zw0. Then A is a noetherian ring of (Krull) dimension one. For such an A we show (Theorem 7.8) that det: ko(A[T]) Pic (A[T]) is an isomorphism. In ?9 we prove also a non-stable analogue of this for T of rank <2. In ?8 we show (Theorem 8.1) that

Weakly injective and weakly projective modules, which are extensions of the classes of quasi-injective modules and quasi-projective modules respectively, are studied. Descriptions are obtained for weakly injective and weakly projective Abelian groups, as well as for some classes of Abelian groups closely related to them.

In this chapter, we establish some basic facts on Clifford theory for induced projective representations. We follow quite closely the exposition for the case of ordinary representations in Chap. 2 and we give the first applications of the theory. Another fundamental application is in Sect. 10.2 on irreducible projective representations of finite Abelian groups.

Units in regular abelian p-group rings

Let G be a finite Abelian group and S be a subset of G. The Cayley sum graph Cay+(G, S) of G with respect to S is a graph whose vertex set is G and two vertices g and h are joined by an edge if and only if g + h є S. In this paper, we prove some basic facts on the domination and total domination numbers of Cayley sum graphs. Then, we find the sharp bounds for domination number of Cay+(Zn, S), where S = {1, 2,..., k} and
, k are positive integers with 1 ≤ k ≤ (n - 1)/2.

Degenerations and orbits in finite abelian groups