Elementary duality of modules

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.