kernelement

Kernelement, or element of the kernel, is a term used in algebra to describe an element that is mapped to the identity element by a homomorphism or linear map. The kernel itself is the set of all such elements, and its specific identity depends on the algebraic structure in question: zero in additive contexts and the multiplicative identity in some multiplicative contexts.

In linear algebra, the kernel of a linear map f: V → W is defined as Ker(f) = {v ∈ V | f(v) = 0}. This set is always a subspace of V, and its dimension is called the nullity of f. The rank-nullity theorem relates the dimension of the kernel to the rank of the map: dim(V) = rank(f) + nullity(f). Elements of the kernel are precisely the vectors that map to the zero vector, and they measure the non-injectivity of the map.

In group theory, for a group homomorphism φ: G → H, the kernel is Ker(φ) = {g ∈ G | φ(g) = e_H}, where e_H is the identity in H. Ker(φ) is a normal subgroup of G, and the quotient group G/Ker(φ) is isomorphic to the image of φ, illustrating how kernels relate to injectivity and structure preservation.

In ring theory and other algebraic settings, kernels are similarly defined using the appropriate identity element (often the zero element in additive structures or the multiplicative identity in certain rings) and retain properties analogous to the linear and group cases.

Computationally, finding a kernelement typically involves solving a homogeneous system of equations (for matrices) or determining elements that map to the identity under the given homomorphism.