Definition:Kernel (Abstract Algebra)

Kernel of Group Homomorphism
Let $\phi: \left({G, \circ}\right) \to \left({H, *}\right)$ be a group homomorphism.

The kernel of $\phi$ is the subset of the domain of $G$ defined as:
 * $\ker \left({\phi}\right) = \left\{{x \in G: \phi \left({x}\right) = e_H}\right\}$

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

That is, $\ker \left({\phi}\right)$ is the subset of $G$ that maps to the identity of $H$.

Kernel of Ring Homomorphism
Let $\left({R_1, +_1, \circ_1}\right)$ and $\left({R_2, +_2, \circ_2}\right)$ be rings.

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

The kernel of $\phi$ is the subset of the domain of $R_1$ defined as:
 * $\ker \left({\phi}\right) = \left\{{x \in R_1: \phi \left({x}\right) = 0_{R_2}}\right\}$.

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

That is, $\ker \left({\phi}\right)$ is the subset of $R_1$ that maps to the zero of $R_2$.

Kernel of Linear Transformation
Let $\phi: G \to H$ be a linear transformation where $G$ and $H$ are $R$-modules.

The kernel of $\phi$, denoted $\ker \left({\phi}\right)$, is the subset $\phi^{-1} \left({\left\{{e_H}\right\}}\right)$ of $G$.

Group definition

 * : $\S 7.4$
 * : $\S 12$
 * : $\S 1.10$: Theorem $22$
 * : Chapter $\text{II}$
 * : $\S 47$
 * : $\S 8$: Definition $8.12$

Ring definition

 * : $\S 2.2$
 * : $\S 57$

Linear Transformation definition

 * : $\S 28$