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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Elementary duality of modules
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by Ivo Herzog PDF
Trans. Amer. Math. Soc. 340 (1993), 37-69 Request permission

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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Additional Information
  • © Copyright 1993 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 340 (1993), 37-69
  • MSC: Primary 03C60; Secondary 16D90
  • DOI: https://doi.org/10.1090/S0002-9947-1993-1091706-3
  • MathSciNet review: 1091706