Biclosed relations with respect to torsion theories and almost continuous bifunctors

摘要

We extend the concept of torsion theories on R-Mod, classically described using the relation H defined by Hom(R)(M,N)=0. For each biclosed relation R with respect to H, we define the conglomerate of R-torsion theories. We introduce the concepts of almost continuous (almost cocontinuous) functor, and almost continuous (AC) bifunctor, and we prove that each AC bifunctor induces a biclosed relation with respect to H. We study biclosed relations induced by adjoint pairs of functors on R-Mod, which correspond to R-R-bimodules. In particular, we study biclosed relations induced by bimodules R/I, and specially when R is a semisimple Artinian ring, proving that in this case the set of all biclosed relations is a Boolean lattice of elements.