First isomorphism theorem for modules ($M/\ker f\cong \mathrm{Im}f$)

For a module homomorphism $f:M\to N$, the map $m+\ker f\mapsto f(m)$ is a module isomorphism $M/\ker f\to \mathrm{Im}f$. The formula is well-defined because two representatives differ by an element of the kernel of a module homomorphism exactly when their images agree; the same observation proves injectivity, and surjectivity onto the image follows by its definition.