Mittag-Leffler modules and positive atomicity

is a monomorphism for all collections of right R-modules {Ni : i ∈ I} . Restricting the Ni to certain classes of modules one obtains various relativized concepts. For instance, Goodearl investigated this notion relativized to flat modules. The present study provides a uniform background for such relativizations and thus covers and generalizes, among others, a number of known results about such modules. The point of view taken here allows, on the other hand, a natural strengthening of the concept of Mittag-Leffler module corresponding to various weaker purities, hence stronger notions of pure-projective (via an appropriate generalization of a characterization due