Rüdiger Göbel

Approximations and endomorphism algebras of modules

The category of all modules over a general associative ring is too complex to admit any reasonable classification. Thus, unless the ring is of finite representation type, one must limit attempts at classification to some restricted subcategories of modules. The wild character of the category of all modules, or of one of its subcategories C is often indicated by the presence of a realization theorem, that is, by the fact that any reasonable algebra is isomorphic to the endomorphism algebra of a module from C. This results in the existence of pathological direct sum decompositions and these are generally viewed as obstacles to the classification. Realization theorems have thus become important indicators of the non-classification theory of modules. In order to overcome this problem, approximation theory of modules has been developed over the past few decades. The idea here is to select suitable subcategories C whose modules can be classified, and then to approximate arbitrary modules by ones from C. These approximations are neither unique nor functorial in general, but there is always a rich supply available appropriate to the requirements of various particular applications. Thus approximation theory has developed into an important part of the classification theory of modules. In this monograph the two methods are brought together. First the approximation theory of modules is developed and some of its recent applications, notably to infinite dimensional tilting theory, are presented. Then some prediction principles from set theory are introduced and these become the principal tools in the establishment of appropriate realization theorems. The monograph starts from basic facts and gradually develops the theory towards its present frontiers. It is suitable both for graduate students interested in algebra and for experts in module and representation theory.

COTORSION THEORIES AND SPLITTERS

Let R be a subring of the rationals. We want to investigate self splitting R-modules G that is Ext R(G; G) = 0 holds and follow Schultz [22] to call such modules splitters. Free modules and torsion-free cotorsion modules are classical examples for splitters. Are there others? Answering an open problem by Schultz [22] we will show that there are more splitters, in fact we are able to prescribe their endomorphism R-algebras with a free R-module structure. As a byproduct we are able to answer a problem of Salce [21] showing that all rational cotorsion theories have enough injectives and enough projectives.

Four submodules suffice for realizing algebras over commutative rings

A positive answer is given to the infinite rank four submodules problem. Let λ be an infinite cardinal, and let A be an algebra over the commutative ring R which can be generated by no more than λ elements. Then there exist four R-submodules of the free A-module of rank λ such that the R-endomorphisms leaving the four submodules invariant are precisely the scalar multiplications by the elements of A.

Uncountable cofinalities of permutation groups

A sufficient criterion is found for certain permutation groups to have uncountable strong cofinality, that is, they cannot be expressed as the union of a countable, ascending chain of proper subsets such that and . This is a strong form of uncountable cofinality for , where each is a subgroup of . This basic tool comes from a recent paper by Bergman on generating systems of the infinite symmetric groups, which is discussed in the introduction. The main result is a theorem which can be applied to various classical groups including the symmetric groups and homeomorphism groups of Cantors discontinuum, the rationals, and the irrationals, respectively. They all have uncountable strong cofinality. Thus the result also unifies various known results about cofinalities. A notable example is the group BSym () of all bounded permutations of the rationals which has uncountable cofinality but countable strong cofinality.

Non-deterministic information systems and their domains

Abstract In the theory of denotational semantics of programming languages Dedekind-complete, algebraic partial orders (domains) frequently have been considered since Scotts and Stracheys fundamental work in 1971 (Stoy, 1977). As Scott (1982) showed, these domains can be represented canonically by (deterministic) information systems. However, recently, more complicated constructions (such as power domains) have led to more general domains (Plotkin, 1976; Smyth and Plotkin, 1977; Smyth, 1983). We introduce non-deterministic information systems and establish the representation theorem similar to Scott (1982) for these more general domains. This result will be the basis for solving recursive domain equations.

The kaplansky test problems — an approach via radicals

Abstract The existence of non-free, k-free Abelian groups and modules (over some non-left perfect rings R) having prescribed endomorphism algebra is established within ZFC + ◊ set theory. The principal technique used exploits free resolutions of non-free R-modules X and is similar to that used previously by Griffith and Eklof; much stronger results than have been obtained heretofore are obtained by coding additional information into the module X. As a consequence we can show, inter alia, that the Kaplansky Test Problems have negative answers for strongly ℵ 1 - free Abelian groups of cardinality ℵ 1 in ZFC and assuming the weak Continuum Hypothesis.

On a theorem of Baer, Schreier, and Ulam for permutations

If w = W(Xl ,...) x,) is a word in free variables xi ,..., X, of a group, it leads to complicated (quite often combinatorial or unsolvable) problems to ask for those elements g of a given group G which are expressible in the form g = a?1 ,..., gJ for some g, ,..., g, E G. In the case of commutators w = X, 0 x2 = -1 -1 Xl . x2 . xi x2 we know of finite groups G and g E G’ (= commutator subgroup) such that g f w(g, , g2) for all g, , g, E G; cf. Huppert [12, p. 2581. More general results of this type are to be found in Hall’s lecture notes [ll] or for arbitrary words in Griffith [lo], Rhemtulla [ 151, or Wilson [ 181. This was exploited in Gobel [7, 81. Again, for w = xi o x2 it was shown dually that each element is expressible by commutators in the cases of permutation groups S, , A, (n > 5); cf. Ore [14] and Ito [13]. Analogous results arc known for matrix groups; cf. Clowes and Hirsch [6]. Hence every element of S, is a product of four elements g, 08, from two conjugacy classes gfm and gin. The analogy is true for A, ; cf. HsiiCh’eng-hao [5]. This result was generalized and carried over to the countable case SK0 of all permutations on N by Bertram [3] : (*) Let p be any permutation of SsO with infinite support. Then every permutation of SE0 is a product of four permutations, each conjugate to p. This theorem reflects-in a very strong version-the fact that SX, has only “very few” normal subgroups as known since 1933 from Schreier and Ulam [16]. From Baer’s result [I] we know the Jordan-Holder series in the general case S, of all permutations on a set of cardinality K > K, , which is even unique if this series is finite. Therefore a generalization of (*) for K > X, is to be expected, where, of course, the cardinalities associated with permutations are to be taken in account. Surprizingly, the cardinality 1 s 1 of the support of a permutation scS*, will be sufficient; the support of s is the set of those elements of the underlying set which are not fixpoints of s:

Large indecomposable roots of Ext

Abstract Let D be any p -cotorsion free (abelian) group. We construct a proper class of indecomposable torsion free groups G such that Ext( D , G )=0. This contrasts with the classical case of D= Q where the only indecomposable torsion free cotorsion groups are Q , and the groups of all p -adic integers for each prime p .

Automorphism groups of fields

We consider pairs (K,G) of an infinite field K or a formally real field K and a group G and want to find extension fields F of K with automorphism group G. If K is formally real then we also want F to be formally real and G must be right orderable. Besides showing the existence of the desired extension fields F, we are mainly interested in the question about the smallest possible size of such fields. From some combinatorial tools, like Shelah’s Black Box, we inherit jumps in cardinalities of K and F respectively. For this reason we apply different methods in constructing fields F: We use a recent theorem on realizations of group rings as endomorphism rings in the category of free modules with distinguished submodules. Fortunately this theorem remains valid without cardinal jumps. In our main result (Theorem 1) we will show that for a large class of fields the desired result holds for extension fields of equal cardinality.

Outer automorphism groups of metabelian groups

Abstract Recall that the outer automorphism group of a group G , denoted Out G , is the quotient group Aut G/ Inn G . If M is any group, then there exists a torsion-free, metabelian group G with trivial center such that Out G≅M . This answers a problem in the Kourovka Notebook (Mazurov, Khukhro, Unsolved problems in group theory; the Kourovka Notebook, Russian Academy of Science, Novosibirsk, 1992).