# Difference between revisions of "Projective covering"

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− | ''of a module'' | + | ''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: 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 [[#References|[a1]]] (cf. also [[Perfect ring]]). | |

− | These notions are completely categorical. A | + | These notions are completely categorical. A [[Grothendieck category]] (also called an $AB5$ category with [[Generator of a category|generator]]s) has the property that injective envelopes always exist. |

====References==== | ====References==== | ||

− | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> H. Bass, "Finitistic homological dimension and a homological generalization of semi-primary rings" ''Trans. Amer. Math. Soc.'' , '''95''' (1960) pp. 466–488</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> N. Popescu, "Abelian categories with applications to rings and modules" , Acad. Press (1973) pp. Sect. 3.10</TD></TR></table> | + | <table> |

+ | <TR><TD valign="top">[a1]</TD> <TD valign="top"> H. Bass, "Finitistic homological dimension and a homological generalization of semi-primary rings" ''Trans. Amer. Math. Soc.'' , '''95''' (1960) pp. 466–488</TD></TR> | ||

+ | <TR><TD valign="top">[a2]</TD> <TD valign="top"> N. Popescu, "Abelian categories with applications to rings and modules" , Acad. Press (1973) pp. Sect. 3.10</TD></TR> | ||

+ | <TR><TD valign="top">[b1]</TD> <TD valign="top"> F.W. Anderson, K.R. Fuller, "Rings and Categories of Modules" Graduate Texts in Mathematics '''13''' Springer (2012) ISBN 1468499130</TD></TR> | ||

+ | </table> | ||

+ | |||

+ | {{TEX|done}} |

## Latest revision as of 20:19, 30 October 2016

*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: 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:**

Projective covering.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Projective_covering&oldid=16493