Elements of Module with Equal Images under Linear Transformations form Submodule

Theorem
Let $G$ and $H$ be $R$-modules.

Let $\phi$ and $\psi$ be linear transformations from $G$ into $H$.

Then the set $S = \set {x \in G: \map \phi x = \map \psi x}$ is a submodule of $G$.

Proof
Let $x, y \in S$.

Let $\lambda \in R$.

Then:

Hence $x + y, \lambda \circ x \in S$.

By Submodule Test, $S$ is a submodule of $G$.

Also see

 * Elements of Group with Equal Images under Homomorphisms form Subgroup