A duality principle for lattices and categories of modules

Abstract

Let R be a ring with 1, R op the opposite ring, and R - Mod the category of left unitary R -modules and R -linear maps. A characterization of well-powered abelian categories A such that there exists an exact embedding functor A → R - Mod is given. Using this characterization and abelian category duality, the following duality principles can be established. Theorem. There exists an exact embedding functor A → R - Mod if and only if there exists an exact embedding functor A op→ R op- Mod. Corollary. If R - Mod has a specified diagram-chasing property, then R op- Mod has the dual property. A lattice L is representable by R -modules if it is embeddable in the lattice of submodules of some unitary left R -module; L ( R ) denotes the quasivariety of all lattices representable by R -modules. Theorem. A lattice L is representable by R -modules if and only if its order dual L ∗ is representable by R op-modules. That is, L( R op)={L ∗:Lϵ L( R)} . If R is a commutative ring with 1 and a specified diagram-chasing result is satisfied in R - Mod, then the dual result is also satisfied in R - Mod. Furthermore, L( R) is self-dual: L( R)= {L ∗:Lϵ L( R)}.