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Model theory of modules. (English) Zbl 0593.16019
This paper was a break-through, both in methods and in results. Exemplifying the strong correlation between model theory and algebra in the context of modules, it had great impetus on further developments of not only model theoretic, but also purely algebraic nature, as well as on those where both aspects overlap essentially (cf. e.g. [A. Pillay and M. Prest, Modules and Stability Theory (preprint 1984)], [A. Facchini, Pac. J. Math. 116, 25-37 (1985; Zbl 0523.16019)], and [M. Prest, Model theory and representation type of algebras, to appear in Proc. Logic Coll. ’86, North Holland], resp.). These two aspects are amalgamated over the concept of pure-injective (p.i.) module (this was apparent already in the work of Eklof and Fisher; here and below consult the paper under review for references). The concept itself reflects the afore-mentioned interaction, for it can be defined algebraically in terms of morphisms as well as (first order) model theoretically as a certain kind of (positively) saturated structure (if the language chosen is the ”right” one, i.e. that containing 0, $$+$$, and unary function symbols for every element of the ring $$R$$). See [B. Wȩglorz, Fundam. Math. 59, 289-298 (1966; Zbl 0221.02039)] for a treatment along these lines in an even much more general context. Incidently, the latter paper is the first I know of where the above language was suggested for model theoretic investigations of modules. Talking of the old days, one should notice it was in another paper of the same series, in [J. Mycielski and C. Ryll-Nardzewski, Fundam. Math. 61, 271-281 (1968; Zbl 0263.08001)], that pure injectivity was established for $$(| R| +\aleph_ 0)^+$$-saturated modules. Due to this fact p.i.’s are so useful in model theoretic respect, for it yields plenty of p.i. models for any theory of $$R$$-modules.
Concerning the author’s methods, in my opinion the main novelty is the introduction, in Section 4, of an appropriate topology on the set of (isomorphism types of) indecomposable p.i.’s and the extensive study of the resulting space, which turns out to be compact (Theorem 4.9), not in general Hausdorff though. Another useful feature of that space is that its closed subsets correspond one-to-one to the complete theories of $$R$$-modules, so that one can talk of the space corresponding to (the complete theory of) a given $$R$$-module (Cor. 4.10). Thus Section 4 could be called the heart of the paper. Although Sections 6, 7, and 8 contain further fundamental investigations related to that space and thus form the stock of methods available for application in Sections 5, 9, 10, and 11, I will not go further into medicine by naming these after other organs (judging from what I heard I should rather like to leave this to Steven Garavaglia instead, who was, by the way, one of the pioneers in the field). (In the light of the above division, the actual partition of the paper into chapters seems somewhat arbitrary to me.)
An important result, which can be considered rather algebraic, is a sufficient and, provided the ring is countable, also necessary condition for a p.i. module to be the p.i. hull of a direct sum of (p.i.) indecomposables (Section 7; whether this condition is necessary in general is still open; a partial answer was given by Ian Hodkinson, Queen Mary College). Note that – even though some decomposition always exists, according to an unpublished result of Fisher’s (cf. Section 6 for a proof) – in general a (p.i.) summand can occur which has no indecomposable direct summands (we call these continuous; if it does not occur the module is called discrete).
A special case of this is an earlier result due to Garavaglia proving discreteness of p.i. modules having so-called elementary Krull dimension (which differs from the usual one in that it refers to the lattice of certain definable subgroups rather than to that of submodules). Examples of those are $$\Sigma$$-p.i. (= totally transcendental) modules and any module over a commutative noetherian ring all of whose localizations at maximal ideals are fields or discrete valuation rings (Cor. 8.2). (Notice that in Theorem 5.2 all the indecomposable p.i.’s are determined over such rings, even without the ”noetherian” assumption.)
One of the main applications of the study of the space of indecomposable p.i.’s is that, given a module M with elementary Krull dimension, one gets a full list of p.i.’s elementarily equivalent to M when taking direct sums of members of that space in a well-prescribed and simple fashion and the pure hull of those afterwards (Theorem 9.1). An immediate consequence is the existence of a smallest p.i., elementarily equivalent to $$M$$ (a ”minimal and prime p.i. model”, Cor. 9.2).
This enables the author to fully describe the uncountable spectrum, i.e. the possible numbers of isomorphism types of models in a given uncountable power of a complete theory of modules over a countable ring (Theorem 10.1). (This is always one of the main tasks to do in the model theory of a certain class, as well as in general model theory.) A particular instance of this is the characterization of uncountable categoricity (Theorem 10.6) which was asked for in [W. Baur, G. Cherlin, and A. Macintyre, J. Algebra 57, 407-440 (1979; Zbl 0401.03012), p. 436].
The situation is much different in case of the countable spectrum: Although the author proved a number of interesting results and reproved everything which had been known before about it, one of the outstanding problems - Vaught’s conjecture - is still open. This conjecture says that a countable elementary theory having uncountably many countable models must have the maximal number of those (i.e. continuum many). Garavaglia proved it for totally transcendental modules (Theorem 10.3(1); later on it was proved for arbitrary totally transcendental theories in [S. Shelah, L. Harrington, and M. Makkai, Isr. J. Math. 49, 259-280 (1984; Zbl 0584.03021)]). Ziegler proved it in the special case when the ring is Dedekind (Theorem 10.3(2); using some work done by the reviewer the same can be shown in the von Neumann regular case). The most recent result in this direction is a proof for modules of Lascar rank 1 (over arbitrary rings) given by Steve Buechler at Berkeley.
As another important application of his methods the author obtains a sufficient condition for decidability of a theory of modules in terms of an effective list of a base of the corresponding space of indecomposables (Theorem 9.4). This yields, among others, transparent new proofs of older decidability results due to Szmielew and Koslov/Kokorin.
In the last section certain stability theoretic concepts due to Shelah are examined in the context of modules, the main one being that of forking (Theorem 11.1, which was independently proved in [25]). Regarding the other results of this section I must say that those are the only ones of the entire paper which have lost a bit of their actuality – for the simple reason that a lot of work improving them has been done since then (cf. [26], also the paper by Pillay and Prest cited in the beginning, and M. Prest’s text ”Model Theory and Modules” forthcoming at Cambridge Univ. Press in the London Math. Soc. Lecture Note Series; this book will cover most of what has been done in the field).
I have not yet mentioned another point making this paper highly useful (and selfcontained), namely the fact that the author begins with a nice treatment of what can be called the fundamentals of the model theory of modules. He thus provides the reader with earlier results concerning Baur’s quantifier elimination (Section 1), Garavaglia’s stability classification (Section 2), and Warfield’s and Fisher’s theory of p.i. hulls (Section 3).
Of the mostly minor misprints let me mention the following: The reference on p. 157 should read [17,II 3.14 (2)$$\Rightarrow (5)]$$; in Lemma 5.5 $$M$$ should be an $$R$$-module; in (a) of the definition on p. 177 the decomposition should read $$M=K\oplus L$$; there is an overbar missing in 6.8.
I conclude by giving references to papers which meanwhile have appeared in print: S. Garavaglia [10], Notre Dame J. Formal Logic 22, 155-162 (1981; Zbl 0438.03037); A. Pillay [24], Fundam. Math. 121, 125-132 (1984; Zbl 0554.03021); M. Prest [26], J. Symb. Logic 50, 202-219 (1985); their joint [25], Proc. Lond. Math. Soc., III. Ser. 46, 365-384 (1983; Zbl 0509.03018).
Reviewer: Ph.Rothmaler

##### MSC:
 16D50 Injective modules, self-injective associative rings 16P60 Chain conditions on annihilators and summands: Goldie-type conditions 03C60 Model-theoretic algebra 16D80 Other classes of modules and ideals in associative algebras 16-02 Research exposition (monographs, survey articles) pertaining to associative rings and algebras 03C45 Classification theory, stability, and related concepts in model theory 03C35 Categoricity and completeness of theories
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