Let $R$ be a commutative ring. Then polynomial ring $R\left[X\right]$ can be looked at as an $R$-algebra free over a singleton. If $S$ is another $R$-algebra then for any element $s\in S$ there is a unique homomorphims $\phi:R\left[X\right]\rightarrow S$ determined by $X\mapsto s$.
So there is a one-to-one correspondence between $\mathbf{Hom}\left(R\left[X\right],S\right)$ and $S$ and the structure on $S$ as $R$-algebra can be transferred to $\mathbf{Hom}\left(R\left[X\right],S\right)$.
In special case $S=R\left[X\right]$ on $\mathbf{End}\left(R\left[x\right]\right)$ there is also a structure induced by composition of endomorphisms. Now we can go the other way: this structure can be transferred to $R\left[X\right]$.
My question: Will this enrich $R\left[X\right]$, and are there examples of practicizing this?
Working with abelian groups a likewise procedure turns abelian group $\mathbb{Z}$ into ring $\mathbb{Z}$.