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 $\mathfrak{K}$ 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.

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Normal epimorphism. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by M.Sh. Tsalenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article