Costar Modules

co-* s -modules

“…In this case, it is clear that fgd-tl( R U ) ⊆ Refl( R U ) and fg-tl(U S ) ⊆ Refl(U S ). It is also proved that Δ preserves exactness of short exact sequences of modules in both fgd-tl( R U ) and fg-tl(U S ) [3]. Hence, it is also interesting to consider the conditions of preserving the exactness of short exact sequences of modules in Refl( R U ).…”

“…Based on these, many corresponding notions are proposed among which the important modules are progenerators, quasi-progenerators [9], tilting modules [1,13], * -modules [6,8,12,14] and so on. Accordingly, the dual concepts are also studied, such as injective cogenerators, quasi-duality modules [10,11], cotilting modules [2,7], costar modules [3,4], etc. From then on, these modules have attracted great interest.…”

On a Natural Duality Between Grothendieck Categories

“…When = Mod-R and M is a S R -bimodule with S = End R M , then taking U = R, V = S, F = Hom R − S M R and G = Hom S − S M R we obtain Colby and Fuller (2001, Proposition 2.6). Section 2 is obtained by adapting analog ideas, results, and proofs contained in the first two sections of Colby and Fuller (2001). Finally, as an application, in Section 3 we get a rigid graded duality and the notion of graded costar module.…”

“…Recall that M is a tilting module in R-Mod if M is a * -module and M = R-Mod where M is a full subcategory of R-Mod formed by all the submodules of modules in Gen M . In order to obtain a dual notion of * -modules, in Colby and Fuller (2001) introduced between the category fgd-tl M R of M-torsionless right R-modules X with Hom R X M R finitely generated over S and the category fg-tl S M of M-torsionless finitely generated left S-modules. In Colby and Fuller (2001) costar modules are characterized, and it is shown that the class of costar modules includes cotilting modules.…”

Coplexes in abelian categories

“…Also, is very useful to generalize such dualities, between module categories, to dualities induced by a pair of adjoint functors between abelian (or, Grothendieck) categories, because they could be applied to different pairs of adjoint functors. In [7], Castaño-Iglesias generalized the notion of costar module, introduced by Colby and Fuller in [8], to the notion of costar object in Grothedieck categories. In [5], the authors extends the notion of f -cotilting module (see, for example, [16]) to the notion of f -cotilting pair of contravariant functors.…”