Elementary duality of modules

Author:
Ivo Herzog

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

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Abstract: Let be a ring. A formula in the language of left -modules is called a positive primitive formula (ppf) if it is of the form where and are matrices of appropriate size with entries in . We apply Prest's notion of , the ppf in the language of right -modules dual to , to show that the model theory of left -modules as developed by Ziegler [Z] is in some sense dual to the model theory of right -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 between the left and right Ziegler spectra which assigns to a given pure-injective indecomposable left -module the dual pure-injective indecomposable right -module . Theorem 6.6 asserts that given a complete theory of left -modules, there is a dual complete theory of right -modules with corresponding Baur-Garavaglia-Monk invariants. In the end, we give some conditions on a pure-injective indecomposable which ensure that its dual may be represented as a hom set of the form where is some ring making into a bimodule and is injective.

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DOI:
https://doi.org/10.1090/S0002-9947-1993-1091706-3

Article copyright:
© Copyright 1993
American Mathematical Society