First isomorphism theorem for modules

First isomorphism theorem (modules): Let $f\colon M\to N$ be a module homomorphism. Then the induced map

is an isomorphism of $R$-modules. Here $\ker(f)$ is the kernel and $\mathrm{im}(f)$ is the image, and $M/\ker(f)$ is a quotient module.

This theorem identifies the “effective domain” of a map with its image and is the basic mechanism behind many results in the theory of exact sequences; it is the module analogue of the first isomorphism theorem for rings