Category:Kernels (Abstract Algebra)

This category contains results about kernels in the context of abstract algebra.
Definitions specific to this category can be found in Definitions/Kernels (Abstract Algebra).

Kernel of Magma Homomorphism

Let $\struct {S, \circ}$ be a magma.

Let $\struct {T, *}$ be an algebraic structure with an identity element $e$.

Let $\phi: \struct {S, \circ} \to \struct {T, *}$ be a homomorphism.

The kernel of $\phi$ is the subset of the domain of $\phi$ defined as:

$\map \ker \phi = \set {x \in S: \map \phi x = e}$

That is, $\map \ker \phi$ is the subset of $S$ that maps to the identity of $T$.

Kernel of Group Homomorphism

Let $\struct {G, \circ}$ and $\struct {H, *}$ be groups.

Let $\phi: \struct {G, \circ} \to \struct {H, *}$ be a group homomorphism.

The kernel of $\phi$ is the subset of the domain of $\phi$ defined as:

$\map \ker \phi := \phi^{-1} \sqbrk {e_H} = \set {x \in G: \map \phi x = e_H}$

where $e_H$ is the identity of $H$.

That is, $\map \ker \phi$ is the subset of $G$ that maps to the identity of $H$.

Kernel of Ring Homomorphism

Let $\struct {R_1, +_1, \circ_1}$ and $\struct {R_2, +_2, \circ_2}$ be rings.

Let $\phi: \struct {R_1, +_1, \circ_1} \to \struct {R_2, +_2, \circ_2}$ be a ring homomorphism.

The kernel of $\phi$ is the subset of the domain of $\phi$ defined as:

$\map \ker \phi = \set {x \in R_1: \map \phi x = 0_{R_2} }$

where $0_{R_2}$ is the zero of $R_2$.

That is, $\map \ker \phi$ is the subset of $R_1$ that maps to the zero of $R_2$.

From Ring Homomorphism Preserves Zero it follows that $0_{R_1} \in \map \ker \phi$ where $0_{R_1}$ is the zero of $R_1$.