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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Elementary duality of modules

Author: Ivo Herzog
Journal: Trans. Amer. Math. Soc. 340 (1993), 37-69
MSC: Primary 03C60; Secondary 16D90
MathSciNet review: 1091706
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Abstract: Let $ R$ be a ring. A formula $ \varphi ({\mathbf{x}})$ in the language of left $ R$-modules is called a positive primitive formula (ppf) if it is of the form $ \exists {\mathbf{y}}\left({AB} \right)\left(\begin{array}{*{20}{c}}x\\ y\\ \end{array} \right) = 0$ where $ A$ and $ B$ are matrices of appropriate size with entries in $ R$. We apply Prest's notion of $ D\varphi ({\mathbf{x}})$, the ppf in the language of right $ R$-modules dual to $ \varphi $, to show that the model theory of left $ R$-modules as developed by Ziegler [Z] is in some sense dual to the model theory of right $ R$-modules. We prove that the topologies on the left and right Ziegler spectra are "isomorphic" (Proposition 4.4). When the lattice of ppfs is well behaved, there is a homeomorphism $ D$ between the left and right Ziegler spectra which assigns to a given pure-injective indecomposable left $ R$-module $ U$ the dual pure-injective indecomposable right $ R$-module $ DU$. Theorem 6.6 asserts that given a complete theory $ T$ of left $ R$-modules, there is a dual complete theory $ DT$ of right $ R$-modules with corresponding Baur-Garavaglia-Monk invariants. In the end, we give some conditions on a pure-injective indecomposable $ _RU$ which ensure that its dual $ DU$ may be represented as a hom set of the form $ {\operatorname{Hom}_S}{(_R}{U_S},{E_S})$ where $ S$ is some ring making $ _R{U_S}$ into a bimodule and $ {E_S}$ is injective.

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Article copyright: © Copyright 1993 American Mathematical Society