Let be a ring with 1. Let and be left -modules, and let be an -module homomorphism. Prove that and are -submodules of and , respectively.
We use the Submodule Criterion for both sets.
Note that , so that and are both nonempty.
Let and let . Note that , so that . Thus is an -submodule of .
Let and let . Note that ; in particular, since and are arbitrary in , by the Submodule criterion we have that is an -submodule of .