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Elementary duality of modules. (English) Zbl 0815.16002
The duality that is the topic of this paper is one which connects the model theories of right and left modules over a given ring. It was defined at the level of formulas by the reviewer [M. Prest, J. Lond. Math. Soc., II. Ser. 38, 403-409 (1988; Zbl 0674.16019)] and, at this level, it can also be discerned, in rather different form, in earlier work of L. Gruson and C. U. Jensen [Lect. Notes Math. 867, 234-294 (1981; Zbl 0505.18005)] (also see [B. Zimmermann- Huisgen and W. Zimmermann, Trans. Am. Math. Soc. 320, 695-713 (1990; Zbl 0699.16019)]).
In Herzog’s paper, this duality is extended considerably and put to many uses. For example, the author shows that the right and left Ziegler spectra over a ring are essentially homeomorphic and that there is a bijection between theories of right and left $$R$$-modules. He shows that a wide class of indecomposable pure-injectives (= points of the Ziegler spectrum) have well-defined duals.
The paper contains interesting and sometimes startling results on a variety of special and general topics. These include modules over Dedekind domains, localisation at a closed subset of the Ziegler spectrum, strongly minimal and $$pp$$-simple indecomposables, local purity, the relation between the dual $$DU$$ of an indecomposable pure-injective $$U$$ and $$\operatorname{Hom}(U,DU\otimes U)$$, duality between flat and absolutely pure modules, totally transcendental modules, Morita duality and elementary duality versus Hom-duality.
The paper is well written and the background material is summarised. An early, key, lemma is a reformulation of the usual criterion for a tensor to be zero: if $$\overline a$$ is an $$n$$-tuple from a left $$R$$-module $$M$$ and if $$\overline c$$ is an $$n$$-tuple from a right $$R$$-module $$N$$ then $$\overline a\otimes\overline c=$$ (that is, $$\sum a_ i\otimes c_ i = 0$$) iff there is some $$pp$$ formula $$\varphi$$ for left modules such that $$M$$ satisfies $$\varphi(\overline a)$$ and $$N$$ satisfies $$D\varphi(\overline c)$$, where $$D \varphi$$ denotes the dual of $$\varphi$$. In many parts of the paper the author works in the context of a category which he denotes $$(R\text{-Mod})^{\text{eq}}$$ – this can be thought of as “positive-$$^{\text{eq}}$$” of the incomplete theory of left $$R$$-modules, with $$pp$$-defined sorts for objects and $$pp$$-definable morphisms between sorts. This category, which is a very natural one for the study of modules using model-theoretic ideas, is, in fact, equivalent to the category ($$R\text{-mod},Ab)^{fp}$$ of finitely presented additive functors from the category of finitely presented $$R$$-modules to the category of abelian groups. The ideas and techniques of this paper have already found many uses and are sure to find many more.

##### MSC:
 16B70 Applications of logic in associative algebras 16D90 Module categories in associative algebras 03C60 Model-theoretic algebra 16D40 Free, projective, and flat modules and ideals in associative algebras
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##### References:
 [1] Paul Eklof and Gabriel Sabbagh, Model-completions and modules, Ann. Math. Logic 2 (1970/1971), no. 3, 251 – 295. · Zbl 0227.02029 [2] Ivo Herzog, Some model theory of modules, Doctoral Dissertation, Univ. Notre Dame, 1989. · Zbl 0898.03014 [3] Carl Faith, Algebra. II, Springer-Verlag, Berlin-New York, 1976. Ring theory; Grundlehren der Mathematischen Wissenschaften, No. 191. · Zbl 0335.16002 [4] Mike Prest, Model theory and modules, London Mathematical Society Lecture Note Series, vol. 130, Cambridge University Press, Cambridge, 1988. · Zbl 0634.03025 [5] Gabriel Sabbagh and Paul Eklof, Definability problems for modules and rings, J. Symbolic Logic 36 (1971), 623 – 649. · Zbl 0251.02052 [6] Bo Stenström, Rings of quotients, Springer-Verlag, New York-Heidelberg, 1975. Die Grundlehren der Mathematischen Wissenschaften, Band 217; An introduction to methods of ring theory. · Zbl 0296.16001 [7] Martin Ziegler, Model theory of modules, Ann. Pure Appl. Logic 26 (1984), no. 2, 149 – 213. · Zbl 0593.16019 [8] Birge Zimmermann-Huisgen and Wolfgang Zimmermann, On the sparsity of representations of rings of pure global dimension zero, Trans. Amer. Math. Soc. 320 (1990), no. 2, 695 – 711. · Zbl 0699.16019
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