\begin{definition}[Definition:Kernel of Linear Transformation]
Let $\phi: G \to H$ be a linear transformation where $G$ and $H$ are $R$-modules.
Let $e_H$ be the identity of $H$.
The '''kernel''' of $\phi$ is defined as:
:$\map \ker \phi := \phi^{-1} \sqbrk {\set {e_H} }$
where $\phi^{-1} \sqbrk S$ denotes the preimage of $S$ under $\phi$.
\end{definition}