Kernel (algebra)
In abstract algebra, the kernel of a homomorphism is a fundamental concept that measures the extent to which the homomorphism fails to be injective (one-to-one). More precisely, the kernel is the set of elements in the domain of the homomorphism that are mapped to the identity element in the codomain.
Formally, let f: G → H be a homomorphism between two algebraic structures G and H. The structures could be groups, rings, vector spaces, modules, or any other structure where the notion of a homomorphism and an identity element is defined. The kernel of f, denoted ker(f), is defined as:
ker(f) = { x ∈ G | f(x) = eH }
where eH is the identity element of H.
Properties and Significance:
- Substructure: The kernel of a homomorphism is always a substructure of the domain G. Specifically:
- If f: G → H is a group homomorphism, then ker(f) is a normal subgroup of G.
- If f: R → S is a ring homomorphism, then ker(f) is an ideal of R.
- If f: V → W is a linear transformation between vector spaces, then ker(f) is a subspace of V.
- Injectivity: A homomorphism f is injective if and only if its kernel is trivial, meaning ker(f) = { eG }, where eG is the identity element of G. This provides a direct link between the kernel and the injectivity of the homomorphism.
- Quotient Structures: The kernel plays a crucial role in the construction of quotient structures and the fundamental isomorphism theorems. For example, in group theory, the first isomorphism theorem states that G / ker(f) is isomorphic to im(f), the image of f. This shows how the kernel "measures" what is lost when G is mapped to H by f.
- Linear Algebra applications: In linear algebra, the kernel of a linear transformation is often referred to as the null space. The dimension of the kernel is called the nullity of the transformation.
- Understanding Homomorphisms: The kernel provides valuable insights into the behavior and structure of homomorphisms. By examining the kernel, one can understand which elements are "identified" by the homomorphism, leading to a better understanding of how the domain G is transformed into the codomain H.
In essence, the kernel is a foundational concept that reveals important information about the structure-preserving maps between algebraic objects and offers a powerful tool for analyzing and understanding these maps.