Definition:Kernel of Ring Homomorphism

From ProofWiki

Jump to: navigation, search

Definition

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 $\phi$ 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 $R_2$.

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