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A morphism having the characteristic property of an imbedding of a group (ring) into a group (ring) as a normal subgroup (ideal). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067550/n0675501.png" /> be a [[Category|category]] with zero morphisms. A morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067550/n0675502.png" /> is called a normal monomorphism if every morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067550/n0675503.png" /> for which it always follows from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067550/n0675504.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067550/n0675505.png" />, that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067550/n0675506.png" />, can be uniquely represented in the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067550/n0675507.png" />. The kernel of any morphism (cf. [[Kernel of a morphism in a category|Kernel of a morphism in a category]]) is a normal monomorphism. The converse is not true, in general; however, if cokernels (cf. [[Cokernel|Cokernel]]) of morphisms exist in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067550/n0675508.png" />, then every normal monomorphism turns out to be the kernel of its cokernel. In an [[Abelian category|Abelian category]] every monomorphism is normal. The concept of a normal monomorphism is dual to that of a [[Normal epimorphism|normal epimorphism]].
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A morphism having the characteristic property of an imbedding of a group (ring) into a group (ring) as a normal subgroup (ideal). Let $\mathfrak{K}$ be a [[Category|category]] with zero morphisms. A morphism $\mu : U \to A$ is called a normal monomorphism if every morphism $\phi : X \to A$ for which it always follows from $\mu \, \alpha = 0$, $\alpha : A \to Y$, that $\phi \, \alpha = 0$, can be uniquely represented in the form $\phi = \phi ' \mu$. The kernel of any morphism (cf. [[Kernel of a morphism in a category|Kernel of a morphism in a category]]) is a normal monomorphism. The converse is not true, in general; however, if cokernels (cf. [[Cokernel|Cokernel]]) of morphisms exist in $\mathfrak{K}$, then every normal monomorphism turns out to be the kernel of its cokernel. In an [[Abelian category|Abelian category]] every monomorphism is normal. The concept of a normal monomorphism is dual to that of a [[Normal epimorphism|normal epimorphism]].
  
  
  
 
====Comments====
 
====Comments====
The above definition is not entirely standard: many authors would define a normal monomorphism to be a morphism which occurs as a kernel. (The term normal subobject is also in use, for an isomorphism class of normal monomorphisms.) In categories without zero morphisms, a substitute for normal monomorphisms is provided by regular monomorphisms, which are those morphisms which occur as equalizers (cf. [[Kernel of a morphism in a category|Kernel of a morphism in a category]]). Every normal monomorphism is regular, but not conversely: in the category of all groups every injective homomorphism is a regular monomorphism, but a normal monomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067550/n0675509.png" /> is an isomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067550/n06755010.png" /> onto a normal subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067550/n06755011.png" />. However, in an additive category the concepts of normal monomorphism and regular monomorphism coincide.
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The above definition is not entirely standard: many authors would define a normal monomorphism to be a morphism which occurs as a kernel. (The term normal subobject is also in use, for an isomorphism class of normal monomorphisms.) In categories without zero morphisms, a substitute for normal monomorphisms is provided by regular monomorphisms, which are those morphisms which occur as equalizers (cf. [[Kernel of a morphism in a category|Kernel of a morphism in a category]]). Every normal monomorphism is regular, but not conversely: in the category of all groups every injective homomorphism is a regular monomorphism, but a normal monomorphism $G \to H$ is an isomorphism of $G$ onto a normal subgroup of $H$. However, in an additive category the concepts of normal monomorphism and regular monomorphism coincide.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  B. Mitchell,  "Theory of categories" , Acad. Press  (1965)  pp. Sect. I.14</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  B. Mitchell,  "Theory of categories" , Acad. Press  (1965)  pp. Sect. I.14</TD></TR></table>
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Latest revision as of 02:26, 14 January 2017

A morphism having the characteristic property of an imbedding of a group (ring) into a group (ring) as a normal subgroup (ideal). Let $\mathfrak{K}$ be a category with zero morphisms. A morphism $\mu : U \to A$ is called a normal monomorphism if every morphism $\phi : X \to A$ for which it always follows from $\mu \, \alpha = 0$, $\alpha : A \to Y$, that $\phi \, \alpha = 0$, can be uniquely represented in the form $\phi = \phi ' \mu$. The kernel of any morphism (cf. Kernel of a morphism in a category) is a normal monomorphism. The converse is not true, in general; however, if cokernels (cf. Cokernel) of morphisms exist in $\mathfrak{K}$, then every normal monomorphism turns out to be the kernel of its cokernel. In an Abelian category every monomorphism is normal. The concept of a normal monomorphism is dual to that of a normal epimorphism.


Comments

The above definition is not entirely standard: many authors would define a normal monomorphism to be a morphism which occurs as a kernel. (The term normal subobject is also in use, for an isomorphism class of normal monomorphisms.) In categories without zero morphisms, a substitute for normal monomorphisms is provided by regular monomorphisms, which are those morphisms which occur as equalizers (cf. Kernel of a morphism in a category). Every normal monomorphism is regular, but not conversely: in the category of all groups every injective homomorphism is a regular monomorphism, but a normal monomorphism $G \to H$ is an isomorphism of $G$ onto a normal subgroup of $H$. However, in an additive category the concepts of normal monomorphism and regular monomorphism coincide.

References

[a1] B. Mitchell, "Theory of categories" , Acad. Press (1965) pp. Sect. I.14
How to Cite This Entry:
Normal monomorphism. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Normal_monomorphism&oldid=40177
This article was adapted from an original article by M.Sh. Tsalenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article