> [!NOTE] **Definition** (Homomorphism between rings with unity) > > Let $(R_{1},+_{R_{1}}, \times_{R_{1}})$ and $(R_{2},+_{R_{2}}, \times_{R_{2}})$ be [[Rings|rings with unity]]. We say that the function $\phi:R_{1}\to R_{2}$ is a *(ring) homomorphism* if $\phi(1_{R_{1}})=1_{R_{2}}$ and it satisfies $\phi(r+_{R_{1}}s)=\phi(r)+_{R_{2}}\phi(s) \text{ and } \phi(r\times_{R_{1}}s)=\phi(r)\times_{R_{2}}\phi(s) $for all $r,s \in R_{1}$. > [!Example] Examples > > [[Homomorphism from ring of polynomials over complex numbers to the complex numbers]]. > > [[Homomorphism from ring of polynomials over reals to the complex numbers]]. > # Properties **Kernel & Image**: The [[Kernel of a Homomorphism of Rings|kernel]] of $\phi,$ denoted $\ker \phi,$ is the preimage of $0_{R_{2}}$ and its [[Image of a Homomorphism of Rings|image]], denoted $\text{Im }\phi,$ is the image of $R_{1}$ under $\phi.$ Note that $\phis [[Kernel of Homomorphism of Rings is an Ideal of Domain|kernel is an ideal]] of $R_{1}$ and that $\phis [[Image of ring homomorphism is a subring|image is a subring]] of $R_{2}.$