Proposition 2

Let A be a ring, M and N two A-modules, and $f$ a nonzero homomorphism from M to N.

a) If M is simple, then $f$ is injective.

b) If N is simple, then $f$ is surjective.

c) If M and N are simple,then $f$ is an isomorphism.

The kernel of $f$ is a proper submodule of M, and the image of $f$ is a nonzero submodule of N.

a) If M is simple, then we have Ker($f$) $= 0$, so $f$ is injective.

b) If N is simple, then we have Im($f$) $= N$, so $f$ is surjective.

c) If M and N are simple, then $f$ is both injective and surjective.