Projective covering

of a left module $M$ over a ring $R$

The dual notion to that of an injective envelope or injective hull. Let $R$ be an associative ring with unit element, $M$ a left module over $R$. From now on, all modules and morphisms are left modules and morphisms of left modules. An epimorphism $q : P \rightarrow M$ is an essential epimorphism if the following holds: $u :P \rightarrow P'$ is an epimorphism if and only if $qu$ is an epimorphism. This is equivalent to saying that $\ker q$ is a superfluous submodule, where $N \subseteq M$ is superfluous if for all submodules $M' \subseteq M$ one has: $M' + N = M$ implies $M' = M$. The notion of an essential epimorphism is dual to that of an essential monomorphism (or essential extension), which is a monomorphism $j : M \rightarrow Q$ such that $v : M' \rightarrow M$ is monomorphic if and only if $jv$ is monomorphic. A projective covering of $M$ is a projective module $P$ together with an essential epimorphism $q : P \rightarrow M$. In contrast to the dual notion of an injective envelope (an injective module $Q$ together with an essential monomorphism $M \rightarrow Q$) projective coverings do not always exist. For instance, indeed especially, projective coverings of Abelian groups ($\mathbf{Z}$-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 Grothendieck category (also called an $AB5$ category with generators) has the property 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