Definition:Kernel (Abstract Algebra)

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

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

$$\mathrm {ker} \left({\phi}\right) = \left\{{x \in S: \phi \left({x}\right) = e_T}\right\}$$

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

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.

Then $$\mathrm {ker} \left({\phi}\right) = \left\{{x \in R_1: \phi \left({x}\right) = 0_{R_2}}\right\}$$.

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