> [!NOTE] Definition (Image 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 [[Image of a set under a function|image]] of $R_{1}$ under $\phi,$ denoted $\text{Im } \phi = \{ \phi(r) \mid r\in R_{1} \}$ is called the image of $\phi.$ # Properties Note that $\phis [[Image of ring homomorphism is a subring|image is a subring]] of $R_{2}.$