Image of a module homomorphism

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

It is a submodule of $N$; see kernel/image are submodules for the standard closure argument.

Images measure surjectivity: $f$ is surjective iff $\operatorname{im}(f)=N$. Together with kernels, images define exactness via the condition $\operatorname{im}(f)=\ker(g)$ for consecutive maps (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 $R$.
• (Edge case) The image of the zero homomorphism is $\{0\}$.