The Torsion Theory Cogenerated by M-Small Modules

Abstract

Let M and N be R-modules. We define

where S denotes the class of all M-small modules. We call N an M-cosingular (non-M-cosingular) module if Z _M (N) = 0 ( Z _M (N) = N). We study the properties of M-cosingular and non-M-cosingular modules in σ[M] We consider the torsion theory cogenerated by M-small modules and show that it is cohereditary when every injective module in σ[M] is amply supplemented. We also give instances where this torsion theory is cohereditary or splits. Finally we characterise lifting modules N ∊ [M] in terms of Z ² _M (N).