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Difference between revisions of "Normal epimorphism"

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A morphism having the characteristic property of the natural mapping of a group onto a quotient group or of a ring onto a quotient ring. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067480/n0674801.png" /> be a [[Category|category]] with zero morphisms. A morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067480/n0674802.png" /> is called a normal epimorphism if every morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067480/n0674803.png" /> for which it always follows from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067480/n0674804.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067480/n0674805.png" />, that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067480/n0674806.png" />, can be uniquely represented in the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067480/n0674807.png" />. The [[Cokernel|cokernel]] of any morphism is a normal epimorphism. The converse assertion is false, in general; however, when morphisms in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067480/n0674808.png" /> have kernels, then every normal epimorphism is a cokernel. In an [[Abelian category|Abelian category]] every epimorphism is normal. The concept of a normal epimorphism is dual to that of a [[Normal monomorphism|normal monomorphism]].
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A [[morphism]] having the characteristic property of the natural mapping of a group onto a [[quotient group]] or of a ring onto a [[quotient ring]]. Let $\mathfrak{K}$ be a [[category]] with zero morphisms. A morphism $\nu : A \rightarrow V$ is called a normal epimorphism if every morphism $\phi : A \rightarrow Y$ for which it always follows from $\alpha.\nu = 0$, $\alpha : X \rightarrow A$, that $\alpha.\phi = 0$, can be uniquely represented in the form $\phi = \nu.\phi'$. The [[cokernel]] of any morphism is a normal epimorphism. The converse assertion is false, in general; however, when morphisms in $\mathfrak{K}$ have [[Kernel of a morphism in a category|kernel]]s, then every normal epimorphism is a cokernel. In an [[Abelian category|Abelian category]] every epimorphism is normal. The concept of a normal epimorphism is dual to that of a [[normal monomorphism]].
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[[Category:Category theory; homological algebra]]

Latest revision as of 21:29, 1 November 2014

A morphism having the characteristic property of the natural mapping of a group onto a quotient group or of a ring onto a quotient ring. Let $\mathfrak{K}$ be a category with zero morphisms. A morphism $\nu : A \rightarrow V$ is called a normal epimorphism if every morphism $\phi : A \rightarrow Y$ for which it always follows from $\alpha.\nu = 0$, $\alpha : X \rightarrow A$, that $\alpha.\phi = 0$, can be uniquely represented in the form $\phi = \nu.\phi'$. The cokernel of any morphism is a normal epimorphism. The converse assertion is false, in general; however, when morphisms in $\mathfrak{K}$ have kernels, then every normal epimorphism is a cokernel. In an Abelian category every epimorphism is normal. The concept of a normal epimorphism is dual to that of a normal monomorphism.

How to Cite This Entry:
Normal epimorphism. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Normal_epimorphism&oldid=15791
This article was adapted from an original article by M.Sh. Tsalenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article