A subset of the domain of a homomorphism consisting of all elements that map to the identity element. The kernel is always a sub-object of appropriate type. E. g. the kernel of a group homomorphism is a subgroup.

of a ring homomorphism It is the preimage of 0 in the codomain of a ring homomorphism. Every ideal is the kernel of a ring homomorphism and vice versa.

of a ring homomorphism  The kernel is the preimage of the element 0 of the codomain of the ring homomorphism. Every ideal is the kernel of a ring homomorphism and vice versa.

In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective.

In mathematics, especially abstract algebra, a kernel is a general construction which measures the failure of a homomorphism or function to be injective.

In category theory and its applications to other branches of mathematics, kernels are a generalization of the kernels of group homomorphisms and the kernels of module homomorphisms and certain other kernels from algebra. Intuitively, the kernel of the morphism f : X → Y is the "most general" morphism k : K → X which, when composed with f, yields zero.