Editing Kernel (algebra) (section)

== Kernel structures ==

The structure of kernels allows for the building of quotient algebras from structures satisfying the properties of kernels. Any [[subgroup]] <math>N</math> of a [[Group (mathematics)|group]] <math>G</math> can construct a quotient <math>G/N</math> by the set of all [[coset]]s of <math>N</math> in <math>G</math>.<ref name="Dummit Group Cosets" /> The natural way to turn this into a group, similar to the treatment for the quotient by a kernel, is to define an operation on (left) cosets by <math>uN \cdot vN = (uv)N</math>, however this operation is well defined [[if and only if]] the subgroup <math>N</math> is closed under [[Conjugation (group action)|conjugation]] under <math>G</math>, that is, if <math>g \in G</math> and <math>n \in N</math>, then <math>gng^{-1} \in N</math>. Furthermore, the operation being well defined is sufficient for the quotient to be a group.<ref name="Dummit Group Cosets" /> Subgroups satisfying this property are called [[normal subgroup]]s.<ref name="Dummit Group Cosets" /> Every kernel of a group is a normal subgroup, and for a given normal subgroup <math>N</math> of a group <math>G</math>, the natural projection <math>\pi(g) = gN</math> is a homomorphism with <math>\ker \pi = N</math>, so the normal subgroups are precisely the subgroups which are kernels.<ref name="Dummit Group Cosets" /> The closure under conjugation, however, gives a criterion for when a subgroup is a kernel for some homomorphism.<ref name="Dummit Group Cosets" />

For a [[Ring (mathematics)|ring]] <math>R</math>, treating it as a group, one can take a quotient group via an arbitrary subgroup <math>I</math> of the ring, which will be normal due to the ring's additive group being [[Abelian group|abelian]]. To define multiplication on <math>R/I</math>, the multiplication of cosets, defined as <math>(r+I)(s+I) = rs + I</math> needs to be well-defined. Taking representative <math>r+\alpha</math> and <math>s+\beta</math> of <math>r + I</math> and <math>s + I</math> respectively, for <math>r,s \in R</math> and <math>\alpha, \beta \in I</math>, yields:<ref name="Dummit Ring Kernels and Ideals" />
: <math>(r + \alpha)(s + \beta) + I = rs + I</math>
Setting <math>r = s = 0</math> implies that <math>I</math> is closed under multiplication, while setting <math>\alpha = s = 0</math> shows that <math>r\beta \in I</math>, that is, <math>I</math> is closed under arbitrary multiplication by elements on the left. Similarly, taking <math>r = \beta = 0</math> implies that <math>I</math> is also closed under multiplication by arbitrary elements on the right.<ref name="Dummit Ring Kernels and Ideals" /> Any subgroup of <math>R</math> that is closed under multiplication by any element of the ring is called an [[Ideal (ring theory)|ideal]].<ref name="Dummit Ring Kernels and Ideals" /> Analogously to normal subgroups, the ideals of a ring are precisely the kernels of homomorphisms.<ref name="Dummit Ring Kernels and Ideals" />