Model Theory Of Modules Research Articles

Simples in a cotilting heart

Every cotilting module over a ring R induces a t-structure with a Grothendieck heart in the derived category D(Mod-R). We determine the simple objects in this heart and their injective envelopes, combining torsion-theoretic aspects with methods from the model theory of modules and Auslander-Reiten theory.

Characterisations of ${\varSigma }$-pure-injectivity in triangulated categories and applications to endocoperfect objects

We provide various ways to characterise $\Sigma $-pure-injective objects in a compactly generated triangulated category. These characterisations mimic analogous well-known results from the model theory of modules. The proof involves two approaches. In

The model theory of modules of a C*-algebra

We study the theory of a Hilbert space H as a module for a unital C*-algebra $${\mathcal{A}}$$ from the point of view of continuous logic. We give an explicit axiomatization for this theory and describe the structure of all the representations which are elementary equivalent to it. Also, we show that this theory has quantifier elimination and we characterize the model companion of the incomplete theory of all non-degenerate representations of $${\mathcal{A}}$$ . Finally, we show that there is an homeomorphism between the space of types of norm less than 1 in this model companion, and the space of quasistates of the C*-algebra $${\mathcal{A}}$$ .

RINGS OF DEFINABLE SCALARS OF VERMA MODULES

Let M be a Verma module over the Lie algebra, sl 2(k), of trace zero 2 × 2 matrices over the algebraically closed field k. We show that the ring, RM, of definable scalars of M is a von Neumann regular ring and that the canonical map from U( sl 2(k)) to RM is an epimorphism of rings. We also describe the Ziegler closure of M. The proofs make use of ideas from the model theory of modules.

EXPLICIT DESCRIPTIONS OF THE INDECOMPOSABLE INJECTIVE MODULES OVER JATEGAONKAR'S RINGS

ABSTRACT In 1968–1969, A. V. Jategaonkar published his famous constructions of left but not right noetherian rings that provided counterexamples to several important conjectures of that era. These examples, and others like them, seemed to indicate that, in general, the task of completely understanding the structure of indecomposable injective modules over one-sided noetherian rings was hopeless. In this paper I show how to deduce by natural methods, directly from the known description of these rings and their properties, explicit computational descriptions of the indecomposable injective left modules over Jategaonkar's rings. I use these explicit descriptions to answer some simple structural questions about the indecomposables. The structure of indecomposable injective modules over a commutative noetherian ring has been well understood since the work of Matlis[10] in the 1950's. There is a natural correspondence between the indecomposable injectives and the localizations of the ring, and the structure of ...

Primitive connected theories

We prove the quantifier-elimination theorem for so-called primitive connected theories, exemplified by theories of modules. The theorem generalizes the well-known Baur-Monk-Garavaglia theorem on the elimination of quantifiers in the model theory of modules. The definition of a class of primitive connected theories, as distinct from modules. is not supposed to impose any conditions on a type of axioms that would specify those theories.

Tensor product and theories of modules

Modules over a ring R, when tensored with an (R, S)-bimodule, are converted to S-modules. Here I consider, from the standpoint of the model theory of modules, the effect of this operation. The primary motivation arises from questions concerning representation type of algebras and interpretability of modules, where such tensor functors play a key role, but this paper is devoted to more general considerations. For instance, the elementary duality of [2] and [1] is generalised here. It is also shown that, although tensor product does not preserve elementary equivalence, one can define the tensor product of two complete theories of R-modules. The results in Section 1 grew out of a number of discussions with T. Kucera.This is our generic situation. We have an (R, S)-bimodule B and we are considering the functor –⊗RBS from Mod – R (the category of right R-modules) to Mod – S which is given on objects by MR ↦ (M ⊗RB)s and has the obvious action on morphisms. There is a somewhat more general situation: namely we may consider the effect of tensoring (T, R)-bimodules over R with B to obtain (T, S)-bimodules. For the intended applications this case is not needed. Moreover, although some results extend to this more general case, there are some which definitely do not (see the example after 2.1). Therefore we confine ourselves to the case first described. We will also consider B itself as a variable and so ask “What is the effect of tensoring R-modules with (R, S)-bimodules?”.

Some model theory over hereditary noetherian domains

Questions in the model theory of modules over hereditary noetherian domains are investigated with particular attention being paid to differential polynomial rings and to generalized Weyl algebras. We prove that there exists no isolated point in the Ziegler spectrum over a simple hereditary generalized Weyl algebra A of the sort considered by Bavula [ Algebra i Analiz 4 (1) (1992), 75–97] over a field k with char( k ) = 0 (the first Weyl algebra A 1 ( k ) is one such) and the category of finite length modules over A does not have any almost split sequence. We show that the theory of all modules over a wide class of generalized Weyl algebras and related rings interprets the word problem for groups, and in the case that the field is countable there exists a superdecomposable pure-injective module over A . This class includes, for example, the universal enveloping algebra Usl 2 ( k ).

An equivalence induced by Ext and Tor applied to the finitistic weak dimension of coherent rings

Let R be a ring, see below for other notation. The functor categories (mod-R, Ab) and ((R-mod)op, Ab) have received considerable attention since the 1960s. The first of these has achieved prominence in the model theory of modules and most particularly in the investigation of the representation theory of Artinian algebras. Both [11, Chapter 12] and [8] contain accounts of the use (mod-R, Ab) may be put to in the model theoretic setting, and Auslander's review, [1], details the application of (mod-R, Ab) to the study of Artinian algebras. The category ((R-mod)op, Ab) has been less fully exploited. Much work, however, has been devoted to the study of the transpose functor between R-mod and mod-R. Warfield's paper, [13], describes this for semiperfect rings, and this duality is an essential component in the construction of almost split sequences over Artinian algebras, see [4]. In comparison, the general case has been neglected. This paper seeks to remedy this situation, giving a concrete description of the resulting equivalence between (mod-R, Ab) and ((R-mod)op, Ab) for an arbitrary ring R.

Four concepts from “geometrical” stability theory in modules

In [H1] Hrushovski introduced a number of ideas concerning the relations between types which have proved to be of importance in stability theory. These relations allow the geometries associated to various types to be connected. In this paper we consider the meaning of these concepts in modules (and more generally in abelian structures). In particular, we provide algebraic characterisations of notions such as hereditary orthogonality, “p -internal” and “p-simple”. These characterisations are in the same spirit as the algebraic characterisations of such concepts as orthogonality and regularity, that have already proved so useful. Of the concepts that we consider, p-simplicity is dealt with in [H3] and the other three concepts in [H2].The descriptions arose out of our desire to develop some intuition for these ideas. We think that our characterisations may well be useful in the same way to others, particularly since our examples are algebraically uncomplicated and so understanding them does not require expertise in the model theory of modules. Furthermore, in view of the increasing importance of these notions, the results themselves are likely to be directly useful in the model-theoretic study of modules and, via abelian structures, in more general stability-theoretic contexts. Finally, some of our characterisations suggest that these ideas may be relevant to the algebraic problem of understanding the structure of indecomposable injective modules.

Imaginary modules

It was a fundamental insight of Shelah that the equivalence classes of a definable equivalence relation on a structure M often behave like (and indeed must be treated like) the elements of the structure itself, and that these so-called imaginary elements are both necessary and sufficient for developing many aspects of stability theory. The expanded structure Meq was introduced to make this insight explicit and manageable. In Meq we have “names” for all things (subsets, relations, functions, etc.) “definable” inside M. Recent results of Bruno Poizat allow a particularly simple and more or less algebraic modification of Meq in the case that M is a module. It is the purpose of this paper to describe this nearly algebraic structure in such a way as to make the usual algebraic tools of the model theory of modules readily available in this more general context. It should be pointed out that some of our discussion has been part of the “folklore” of the subject for some time; it is certainly time to make this “folklore” precise, correct, and readily available.Modules have proved to be good examples of stable structures. Not, we mean, in the sense that they are well-behaved (which, in the main, they are), but in the sense that they are straightforward enough to provide comprehensible illustrations of concepts while, at the same time, they have turned out to be far less atypical than one might have supposed. Indeed, a major feature of recent stability theory has been the ubiquitous appearance of modules or more general “abelian structures” in abstract stable structures.

Modules and Stability Theory

Modules are now widely recognized as important examples of stable structures. In fact, in the light of results and conjectures of Zilber [Zi] (${\aleph _1}$-categorical structures are “field-like”, “module-like” or “trivial”), we may consider modules as one of the typical examples of stable structures. Our aim here is both to prove some new results in the model theory of modules and to highlight the particularly clear form of, and the algebraic content of, the concepts of stability theory when applied to modules. One of the main themes of this paper is the connection between stability-theoretic notions, such as ranks, and algebraic decomposition of models. We will usually work with $T$, a complete theory of $R$-modules, for some ring $R$. In $\S 2$ we show that the various stability-theoretic ranks, when defined, are the same. In $\S 3$ we show that $T$ (not necessarily superstable) is nonmultidimensional (in the sence of Shelah [Sh1]). In $\S 4$ we consider the algebraic content of saturation and we show, for example, that if $M$ is a superstable module then $M$ is $F_{{\aleph _0}}^a$-saturated just if $M$ is pure-injective and realizes all types in finitely many free variables over $\phi$. In $\S 5$ we use our methods to reprove Ziegler’s theorem on the possible spectrum functions. In $\S 6$ we show the profusion (in a variety of senses) of regular types. In $\S 7$ we give a structure theorem for the models of $T$ in the case where $T$ has $U$-rank 1.

The generalised RK-order, orthogonality and regular types for modules

I characterise various model-theoretic properties of types, in complete theories of modules, in terms of the algebraic structure of pure-injective modules. More specifically, I consider the generalised RK-order, and the relation of domination between types, orthogonality of types, and regular types. It will be seen that, essentially, it is the stationary types over pure-injective models which bear the algebraic structural information.For background in the model theory of modules, see [4], [5], [13], [17], [19], and for the model-theoretic background [10], [11], [12], [18]. More specific references will be given in the text. I summarise some principal definitions and results below. First, though, let me describe the main results.Throughout this paper, R will be a ring with 1; the language will be that for (right R-) modules. All types will be complete types in a complete theory, T, of R-modules. The notation is reserved for an extremely saturated model of T, inside which all “small” situations may be found. Thus, sets of parameters are just subsets of , and all models will be elementary substructures of . A positive primitive, or pp, formula is one which is equivalent to a formula of the formwith the rij ∈ R.Suppose that p and q are 1-types over a pure-injective model M.

Rings of finite representation type and modules of finite Morley rank

This paper deals with rings whose modules have very pleasant decomposition properties: the right pure semisimple rings. The category of right modules over such a ring is, in many ways, similar to the category of modules over a semisimple artinian ring. Yet it is still an open question whether or not pure semisimplicity actually is a two-sided property (and hence [30] coincides with being of finite representation type). Most of the arguments in this paper are model-theoretic-algebraic in nature: it seems that ideas from model theory, and from the model theory of modules, find natural application in the context considered here. The paper is organized into three sections. The first is introductory and with it I attempt to make the other sections accessible to a reasonably wide readership. In the second section are gathered together various equivalents to right pure semisimplicity. Their equivalence is given a comparatively short, unified, essentially model-theoretic-algebraic proof. The main theorem, in the third section, is that a ring is of finite representation type if and only if all its (right) modules have finite Morley rank. A number of related results are developed. Most of these involve some model theory in their statement, but many are, in essense, algebraic. Since a good deal of information has been compressed into the introductory section, this section should, perhaps, first be read through quickly, and then referred back to as the need arises. This section also is meant to serve as background to a sequel, in which I consider pp formulas and types in terms of the corresponding matrices, and in which the main aim is more purely algebraic. The global conventions are: R denotes a ring with identity; “module”

Some model theory of modules. III. On infiniteness of sets definable in modules

This is the third and last part of an investigation in several topics of first order model theory of modules which I began in Some model theory of modules. I. On total transcendence of modules (this Journal, vol. 48 (1983), pp. 570–574) and continued in Some model theory of modules. II. On stability and categoricity of flat modules (this Journal, vol. 48 (1983), pp. 970–985). Throughout I refer to these papers as “Part I” and “Part II”.Although these parts are only loosely connected, I will tacitly use the notation and preliminaries already introduced in the preceding ones. Further details are given in §0. Concerning Part II, the reader is assumed to be familiar with most of §1, a very small part of §4, and the criterion for elementary equivalence of modules over regular rings given in §3 (Lemma 20) of that part.Using those tools in this paper I consider completeness of the (elementary) theory of all modules (and of some of its extensions) (§2), and eliminability of cardinality quantifiers in the elementary theories of modules (§3).§2 is almost entirely devoted to a new short proof based on the technique developed in Part II of a theorem of Tukavkin which seems to be the first relevant result concerning the completeness of the theory of all modules (after the simple observation that any theory of infinite vector spaces is complete).

Some model theory of modules. II. on stability and categoricity of flat modules

This is the second part of a study on model theory of modules begun in [RO]. Throughout, I refer to that paper as “Part I”. I observed there a coincidence between some algebraic and logical points of view in the theory of modules, which led to a convenient representation of p.p. definable sets in flat modules (§1); it becomes especially nice in the case of regular rings (cf. Remark 7 in Part I). Using this in the present paper I obtain a simplified criterion for total transcendence and superstability in the case of flat modules. This enables me to give, besides partial results for arbitrary flat modules (§2), a complete description of the stability classes for modules over regular rings (§3). Particularly, it turns out that a totally transcendental module over a regular ring can be regarded as a module over a semisimple ring (cf. Remark 10 in Part I). This is the crucial observation for the examination of categoricity here: By Morley's theorem, a countable ℵ1-categorical theory is totally transcendental. Consequently, an ℵ1-categorical module over a countable regular ring can be regarded as a module over a semisimple ring. That is why I separately treat categoricity of modules over semisimple rings (§4), even without any assumption on the power of the ring. As a consequence I obtain a complete description of ℵ1-categorical modules over countable regular rings (§5). In the investigation presented here the main tool is the technique of idempotents avoiding the commutativity assumption made in the corresponding results of Garavaglia [GA 1, pp. 86–88], who used maximal ideals for that purpose. At the end of the present paper I show how to derive these latter results in our context.On the way I simplify the known criterion for elementary equivalence for modules over regular rings (§3), which simplifies again in case of semisimple rings (§4). Such a criterion is needed in order to construct Vaughtian pairs (in the categoricity consideration) and it turns out to be useful also for another purpose treated in the third part of this series of papers.

Some model theory of modules. I. On total transcendence of modules

This is the first part of a series of papers on certain topics of model theory of modules, which will further consist of Part II. On stability and categoricity of flat modules and Part III. On infiniteness of sets definable in modules. Although these parts are only weakly connected, it is convenient to have common background and notation. The first part was written in Summer 1980, when I did not know about either footnote (3) appearing in the final version of [GA 2] or about the work of the “Bedford College Group” concerning stability theory of modules. I would like to thank Wilfrid Hodges and Mike Prest for their interest in this note, which encouraged me to include it as the first part of the present series, and for acquainting me with their unpublished work, to which I refer in footnotes. I should also like to thank the referee for his suggestions, in particular for correcting the argument in Remark (5) and for drawing my attention to [Z-H].A syntactical analysis of Zimmermann's notion of “endlich matriziellen Untergruppen” of modules [ZI, p. 1087] shows that it coincides with the notion of subgroups definable by positive primitive formulae (without parameters) [GA 1, p. 80]. Then combination of [ZI, Folgerung 3.4] with [GA 1, Lemma 5] (or [GA 2, Theorem 1] for uncountable rings) yields purely algebraic criteria for total transcendence of modules. I present here a proof of this result using only the following facts.(G1) A module M is totally transcendental iff it satisfies the minimal condition on subgroups definable by positive primitive formulae (without parameters) [GA 1, Lemma 5] (and [GA 2, Theorem 1] for uncountable rings).