What is the kernel of a homomorphism?

The kernel of a homomorphism is a fundamental concept in abstract algebra, representing the collection of elements from the domain of a homomorphism that map to the identity element of the codomain. It is essentially the inverse image of the identity element under the homomorphism.

Understanding Homomorphisms

Before diving into the kernel, it's crucial to understand what a homomorphism is. In mathematics, particularly in algebra, a homomorphism is a function between two algebraic structures (like groups, rings, or vector spaces) of the same type that preserves the operations of the structures. This means that if you apply the operation in the domain and then the function, it's the same as applying the function first and then the operation in the codomain.

Defining the Kernel

The kernel, typically denoted as ker(φ) for a homomorphism φ, precisely identifies those elements in the domain that "disappear" or "collapse" into the identity element of the codomain.

• For most algebraic structures (like rings, modules, or vector spaces) where the primary operation is typically written additively, the kernel is the inverse image of the zero element. That is, ker(φ) = {x | φ(x) = 0}, where 0 is the additive identity in the codomain.
• For groups, where the operation can be multiplicative, the kernel is the inverse image of the identity element. If the group operation is denoted multiplicatively, ker(φ) = {g | φ(g) = e'}, where e' is the identity element of the codomain group. If the group operation is denoted additively, it's the inverse image of 0, similar to rings or vector spaces.

In general terms, if φ: A → B is a homomorphism, the kernel of φ is defined as:
ker(φ) = {a ∈ A | φ(a) = e_B}
where e_B is the identity element (zero for additive structures, one for multiplicative groups) in the codomain B.

Importance and Significance of the Kernel

The kernel is far more than just a definition; it's a powerful tool with significant implications in abstract algebra:

• Injectivity Test: A homomorphism φ is injective (one-to-one) if and only if its kernel contains only the identity element of the domain. In other words, φ is injective if and only if ker(φ) = {e_A}.
• Structure Preservation: The kernel itself is not just a subset; it's a special substructure of the domain. For instance:
  • The kernel of a group homomorphism is always a normal subgroup.
  • The kernel of a ring homomorphism is always an ideal.
  • The kernel of a linear map (homomorphism of vector spaces) is always a vector subspace.
• Isomorphism Theorems: The kernel plays a central role in the fundamental Isomorphism Theorems in algebra, which relate quotient structures to images of homomorphisms. For example, the First Isomorphism Theorem states that the image of a homomorphism is isomorphic to the quotient of its domain by its kernel.

Examples of Kernels Across Different Structures

To illustrate, let's look at specific examples:

1. Kernel of a Group Homomorphism

Let φ: G → H be a group homomorphism.

• Definition: ker(φ) = {g ∈ G | φ(g) = e_H}, where e_H is the identity element in group H.
• Example: Consider the homomorphism φ: Z → Z_n (integers under addition to integers modulo n under addition), defined by φ(k) = k mod n.
  • The identity element in Z_n is 0.
  • The elements k ∈ Z that map to 0 mod n are precisely the multiples of n.
  • Therefore, ker(φ) = {..., -2n, -n, 0, n, 2n, ...} = nZ. This forms a normal subgroup of Z.

2. Kernel of a Ring Homomorphism

Let φ: R → S be a ring homomorphism.

• Definition: ker(φ) = {r ∈ R | φ(r) = 0_S}, where 0_S is the additive identity (zero) in ring S.
• Example: Consider the evaluation homomorphism φ: R[x] → R (polynomials with real coefficients to real numbers) defined by φ(p(x)) = p(0).
  • The zero element in R is 0.
  • The elements p(x) ∈ R[x] that map to 0 are the polynomials where p(0) = 0. This means x must be a factor of p(x).
  • Therefore, ker(φ) = {p(x) ∈ R[x] | p(0) = 0} = <x>, which is the ideal generated by x (i.e., all polynomials divisible by x).

3. Kernel of a Linear Map (Vector Space Homomorphism)

An important special case is the kernel of a linear map between vector spaces.
Let T: V → W be a linear map (where V and W are vector spaces over the same field).

• Definition: ker(T) = {v ∈ V | T(v) = 0_W}, where 0_W is the zero vector in W.
• Example: Consider the linear map T: R^3 → R^2 defined by T(x, y, z) = (x, y).
  • The zero vector in R^2 is (0, 0).
  • The vectors (x, y, z) ∈ R^3 that map to (0, 0) are those where x = 0 and y = 0.
  • Therefore, ker(T) = {(0, 0, z) | z ∈ R}. This is the z-axis, which is a subspace of R^3. In linear algebra, the kernel of a linear map is also known as the null space of the transformation.

Summary Table

The kernel serves as a bridge, connecting the structure of the domain, the nature of the homomorphism, and the resulting structure in the codomain.