# Kernel

In general, a kernel is a measure of the failure of a homomorphism to be injective.

In set theory, if $S$ and $T$ are sets, with $f$ a function mapping $S$ into $T$, the kernel of $f$ is quotient set of $S$ under the equivalence relation $R(x,y)$ defined as "$f(x)=f(y)$".

In algebra, a kernel is generally the inverse image of an identity element under a homomorphism. For instance, in group theory, if $G$ and $H$ are groups, and $f : G \to H$ is a homomorphism of groups, the kernel of $f$ is the set of elements of $G$ that map to the identity of $H$, i.e., the set $f^{-1}(e_{H})$. The kernel is a normal subgroup of $G$, and in fact, every normal subgroup of $G$ is the kernel of a homomorphism. Similarly, in ring theory, the kernel of a homomorphism is the inverse image of zero; the kernel is a two-sided ideal of the ring, and every two-sided ideal of a ring is the kernel of a ring homomorphism.