Image of a module homomorphism

Let $f:M\to N$ be a module homomorphism. The image of $f$ is

\operatorname{im}(f)=\{f(m): m\in M\}\subseteq N.

It is a submodule of $N$ .

Properties

The map $f$ is surjective if and only if $\operatorname{im}(f)=N$ . For consecutive module homomorphisms $M\xrightarrow{f}N\xrightarrow{g}P$ , exactness at $N$ means $\operatorname{im}(f)=\ker(g)$ ; see exactness via kernels and images.

Examples

For $f:\mathbb Z\to\mathbb Z$ given by $f(n)=2n$ , the image is $2\mathbb Z$ .
For the projection $\pi:R^2\to R$ , $\pi(a,b)=a$ , the image is all of $R$ .
The image of the zero homomorphism is $\{0\}$ .