Knowlpedia

Image of a ring homomorphism

The subset of the codomain attained by a ring homomorphism.

The image of a φ:R→S\varphi:R\to S is

im⁡(φ)={φ(r):r∈R}⊆S,\operatorname{im}(\varphi)=\{\varphi(r):r\in R\}\subseteq S,

i.e. the of RR under φ\varphi.

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

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