# Difference between revisions of "Normal monomorphism"

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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 | + | 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 | + | 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=18683