The Cotranspose And The Adjoint Cotranspose Of Modules Relative To Subcategories

Let R be a commutative ring,B an R-module and χ be a subcategory of R-modules.The cotransposes and the adjoint cotransposes of modules relative to subcategories are investigated in this paper.Firstly,we prove that the coproperχ(B)-cotransposes of modules are equivalent in different χ-copresentations and an R-module M is an χ(B)-cotranspose of an R-module N if and only if there exists an exact sequence 0→M→cTrBBN→ LB → 0 in R-modules where L∈χ and Tor1R(B,LB)=0.Secondly,we also prove that the proper χ(B)-adjoint cotransposes of modules are equivalent in different χ-presentations and an R-module M is an χ(B)-adjoint cotranspose of an R-module N if and only if there exists an exact sequence 0→B(?)RL→acTrBBN→M→0 where L∈E χ and ExtR1(B,B(?)RL)=0.