Editing Kernel (algebra) (section)

== Quotient algebras ==
The kernel of a homomorphism can be used to define a [[Quotient algebra (universal algebra)|quotient algebra]]. For instance, if <math>\varphi: G \to H </math> denotes a group homomorphism, and denote <math>K = \ker \varphi </math>, then consider <math>G/K</math> to be the set of [[Fiber (mathematics)|fibers]] of the homomorphism <math>\varphi</math>, where a fiber is merely the set of points of the domain mapping to a single chosen point in the range.<ref name="Dummit Group Cosets">{{harvnb|Dummit|Foote|2004|pp=74,76–77,80–82}}</ref> If <math>X_a \in G/K</math> denotes the fiber of the element <math> a \in H </math>, then a group operation on the set of fibers can be endowed by <math>X_a X_b = X_{ab}</math>, and <math>G/K</math> is called the quotient group (or factor group), to be read as "G modulo K" or "G mod K".<ref name="Dummit Group Cosets" /> The terminology arises from the fact that the kernel represents the fiber of the identity element of the range, <math>H</math>, and that the remaining elements are simply "translates" of the kernel, so the quotient group is obtained by "dividing out" by the kernel.<ref name="Dummit Group Cosets" />

The fibers can also be described by looking at the domain relative to the kernel; given <math>X \in G/K</math> and any element <math> u \in X </math>, then <math> X = uK = Ku </math> where:<ref name="Dummit Group Cosets" />
: <math> uK = \{ uk \ | \ k \in K \} </math>
: <math> Ku = \{ ku \ | \ k \in K \} </math>
these sets are called the [[coset|left and right cosets]] respectively, and can be defined in general for any arbitrary [[subgroup]] aside from the kernel.<ref name="Dummit Group Cosets" /><ref>{{harvnb|Hungerford|2014|pp=237–239}}</ref><ref>{{harvnb|Fraleigh|Katz|2003|p=97}}</ref> The group operation can then be defined as <math>uK \circ vK = (uk)K</math>, which is well-defined regardless of the choice of representatives of the fibers.<ref name="Dummit Group Cosets" /><ref>{{harvnb|Fraleigh|Katz|2003|p=138}}</ref>

According to the [[Isomorphism theorems|first isomorphism theorem]], there is an isomorphism <math>\mu: G/K \to \varphi(G)</math>, where the later group is the image of the homomorphism <math>\varphi</math>, and the isomorphism is defined as <math>\mu(uK)=\varphi(u)</math>, and such map is also well-defined.<ref name="Dummit Group FIT" /><ref>{{harvnb|Fraleigh|Katz|2003|p=307}}</ref>

For [[Ring (mathematics)|rings]], [[Module (mathematics)|modules]], and [[vector space]]s, one can define the respective quotient algebras via the underlying additive group structure, with cosets represented as <math>x+K</math>. Ring multiplication can be defined on the quotient algebra the same way as in the group (and be well-defined).<ref name="Dummit Ring Kernels and Ideals"/> For a ring <math>R</math> (possibly a [[Field (mathematics)|field]] when describing vector spaces) and a module homomorphism <math>\varphi: M \to N</math> with kernel <math> K = \ker \varphi </math>, one can define scalar multiplication on <math>G/K</math> by <math>r(x+K)=rx+K</math> for <math>r \in R</math> and <math>x \in M</math>, which will also be well-defined.<ref>{{harvnb|Dummit|Foote|2004|pp=345–349}}</ref>