In general, a kernel is a measure of the failure of a homomorphism to be injective.
In set theory, if and are sets, with a function mapping into , the kernel of is quotient set of under the equivalence relation defined as "".
In algebra, a kernel is generally the inverse image of an identity element under a homomorphism. For instance, in group theory, if and are groups, and is a homomorphism of groups, the kernel of is the set of elements of that map to the identity of , i.e., the set . The kernel is a normal subgroup of , and in fact, every normal subgroup of 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.
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