Projective covering

of a module

This is the dual notion of that of an injective envelope or injective hull (cf. Injective module). Let be an associative ring with unit element, a left module over . From now on, all modules and morphisms are left modules and morphisms of left modules. An epimorphism is an essential epimorphism if the following holds: is an epimorphism if and only if is an epimorphism. This is equivalent to saying that is a superfluous submodule, where is superfluous if for all submodules one has: implies . The notion of an essential epimorphism is dual to that of an essential monomorphism (or essential extension), which is a monomorphism such that is monomorphic if and only if is monomorphic. A projective covering of is a projective module together with an essential epimorphism . In contrast to the dual notion of an injective envelope (an injective module together with an essential monomorphism ) projective coverings do not always exist. For instance, indeed especially, projective coverings of Abelian groups (-modules) do not exist. The rings for which projective coverings of modules do exist have been characterized [a1] (cf. also Perfect ring).

These notions are completely categorical. A so-called category with generators (also called a Grothendieck category) is such that injective envelopes always exist.

References

[a1]	H. Bass, "Finitistic homological dimension and a homological generalization of semi-primary rings" Trans. Amer. Math. Soc. , 95 (1960) pp. 466–488
[a2]	N. Popescu, "Abelian categories with applications to rings and modules" , Acad. Press (1973) pp. Sect. 3.10