Is there an idempotent e in a ring $R$ such that the functor $T_e$, induced by $e$, is not full?

Question

Let $R$ be a ring and let $e$ be a non-zero idempotent in $R$. For each $R$-module $M$, define $T_e: M \rightarrow eM$, where $eM$ is the left $eRe$-module. For each pair of $R$-modules $M, N$ and each left $R$-homomorphism $f: M \rightarrow N$, let $T_e(f): f \mapsto f|_{eM}$. Then $T_e$ defines a covariant additive functor from $R-Mod$ to $eRe-Mod$.

Is there an example of ring $R$ and non-zero idempotent $e$ such that the above functor $T_e$ is not full?

Answer 1

Let $R=\begin{pmatrix}k&k\\0&k\end{pmatrix}$, the ring of upper triangular $2\times2$ matrices over a field $k$, let $e=\begin{pmatrix}1&0\\0&0\end{pmatrix}$, so that $eRe\cong k$, and let $M=R$, so that $eM=\begin{pmatrix}k&k\\0&0\end{pmatrix}$.

Then $\operatorname{Hom}_R(M,M)$ is $3$-dimensional, but $\operatorname{Hom}_{eRe}(eM,eM)$ is $4$-dimensional, so $T_e$ can't be full.