> [!NOTE] Definition (Kernel of ring homomorphism) > Let $R_{1}$ and $R_{2}$ be [[Rings|rings]] and let $\phi:R_{1} \to R_{2}$ be [[Homomorphism of Rings|ring homomorphism]]. Then the [[Preimage (of set under a function)|preimage]] of $0_{R_{2}}$ under $\phi,$ denoted $\ker \phi = \{ r\in R_{1} \mid \phi(r)=0_{R_{2}} \}$ is called the kernel of $\phi.$ # Properties Note that $\phis [[Kernel of Homomorphism of Rings is an Ideal of Domain|kernel is an ideal]] of $R_{1}.$