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 $$G$$.