Biclosed relations with respect to torsion theories and almost continuous bifunctors. Issue 2 (1st February 2017) - Fernández-Alonso, Rogelio; Cerda, Erwin
ABSTRACT: We extend the concept of torsion theories on R -Mod, classically described using the relation ℋ defined by Ho m 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 2 n 2 elements.