Editing Kernel (algebra) (section)

== Universal algebra ==
All the above cases may be unified and generalized in [[universal algebra]]. Let ''A'' and ''B'' be [[algebraic structure]]s of a given type and let ''f'' be a homomorphism of that type from ''A'' to ''B''. Then the ''kernel'' of ''f'' is the subset of the [[direct product]] {{nowrap|''A'' × ''A''}} consisting of all those [[ordered pair]]s of elements of ''A'' whose components are both mapped by ''f'' to the same element in ''B''.<ref>{{harvnb|Burris|Sankappanavar|2012|p=44}}</ref><ref name="McKenzie Kernel"/> The kernel is usually denoted {{nowrap|ker ''f''}} (or a variation). In symbols:
: <math>\operatorname{ker} f = \left\{\left(a, b\right) \in A \times A : f(a) = f\left(b\right)\right\}\mbox{.}</math>

The homomorphism ''f'' is injective if and only if its kernel is exactly the diagonal set {{nowrap|{{mset|(''a'', ''a'') : ''a'' &isin; ''A''}}}}, which is always at least contained inside the kernel.<ref>{{harvnb|Burris|Sankappanavar|2012|p=50}}</ref><ref name="McKenzie Kernel"/>

It is easy to see that ker ''f'' is an [[equivalence relation]] on ''A'', and in fact a [[congruence relation]].
Thus, it makes sense to speak of the [[quotient (universal algebra)|quotient algebra]] {{nowrap|''A'' / (ker ''f'')}}.
The [[isomorphism theorem#General|first isomorphism theorem]] in general universal algebra states that this quotient algebra is naturally isomorphic to the image of ''f'' (which is a [[subalgebra]] of ''B'').<ref>{{harvnb|Burris|Sankappanavar|2012|pp=44–46}}</ref>