In a category of modules the
notions of ρ-injectivity (with respect to a torsion radical ρ) and quasi-injectivity
can be generalized to a notion of injectivity with respect to two preradicals
simultaneously. Using this general definition an analog of Baer’s condition for
injectivity is obtained, as well as other generalizations of results for injective and
quasi-injective modules. An alternate approach (not requiring the existence of
injective envelopes) is given for abelian categories, with the results stated in dual
form for projectivity.
