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''of a module''
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''of a left module $M$ over a ring $R$''
  
This is the dual notion of that of an injective envelope or injective hull (cf. [[Injective module|Injective module]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075200/p0752001.png" /> be an associative ring with unit element, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075200/p0752002.png" /> a left module over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075200/p0752003.png" />. From now on, all modules and morphisms are left modules and morphisms of left modules. An epimorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075200/p0752004.png" /> is an essential epimorphism if the following holds: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075200/p0752005.png" /> is an epimorphism if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075200/p0752006.png" /> is an epimorphism. This is equivalent to saying that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075200/p0752007.png" /> is a superfluous submodule, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075200/p0752008.png" /> is superfluous if for all submodules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075200/p0752009.png" /> one has: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075200/p07520010.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075200/p07520011.png" />. The notion of an essential epimorphism is dual to that of an essential monomorphism (or essential extension), which is a monomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075200/p07520012.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075200/p07520013.png" /> is monomorphic if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075200/p07520014.png" /> is monomorphic. A projective covering of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075200/p07520015.png" /> is a [[Projective module|projective module]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075200/p07520016.png" /> together with an essential epimorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075200/p07520017.png" />. In contrast to the dual notion of an injective envelope (an [[Injective module|injective module]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075200/p07520018.png" /> together with an essential monomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075200/p07520019.png" />) projective coverings do not always exist. For instance, indeed especially, projective coverings of Abelian groups (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075200/p07520020.png" />-modules) do not exist. The rings for which projective coverings of modules do exist have been characterized [[#References|[a1]]] (cf. also [[Perfect ring|Perfect ring]]).
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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 so-called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075200/p07520021.png" /> category with generators (also called a Grothendieck category) is such that injective envelopes always exist.
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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>
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<table>
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<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>
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<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>
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<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>
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</table>
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{{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