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Projective covering

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of a left module 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: this is equivalent to the condition that \mathrm{im}\, v is an essential submodule. 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
[b1] F.W. Anderson, K.R. Fuller, "Rings and Categories of Modules" Graduate Texts in Mathematics 13 Springer (2012) ISBN 1468499130
How to Cite This Entry:
Injective envelope. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Injective_envelope&oldid=39547