Image of a ring homomorphism

The image of a ring homomorphism $\varphi:R\to S$ is

i.e. the image of $R$ under $\varphi$.

The image is a subring of $S$ (and a unital subring if $\varphi$ is unital and $S$ is unital). Together with the kernel, the image governs the structure of $\varphi$ via isomorphism theorems.

Examples:

• For the inclusion $\mathbb Z\hookrightarrow \mathbb Q$, the image is $\mathbb Z\subseteq \mathbb Q$.
• For the reduction map $\mathbb Z\to \mathbb Z/n\mathbb Z$, the image is all of $\mathbb Z/n\mathbb Z$.
• For evaluation $k[x,y]\to k[x]$ at $y=0$, the image is $k[x]$.