How to look at a polynomial ring based on a ring that is not commutative? When I first met polynomial rings $R[X]$ I wondered: 'where do they come from?' Later the idea that - if $R$ is commutative - they could be interpreted as $R$-algebras free over a singleton brought relief. But what if $R$ is not commutative? Then this interpretation seems to stop functioning. If $R$ is not commutative then free over a singleton also demands elements like (for instance) $X^{n}a$ next to the usual $aX^{n}$, isn't that so? Is there another interpretation for them?
 A: This is a great question.  The "free ring over $R$" in one variable in the noncommutative setting is given by $R * \mathbb{Z}[X]$, where the "star" is the free product (= coproduct in the category of rings).  It is universal among all rings $S$ that are equipped with two ring homomorphisms $R \to S$ and $\mathbb{Z}[X] \to S$.
On the other hand, the ring $R[X]$ is isomorphic to $R \otimes \mathbb{Z}[X]$.  It is universal among all rings $S$ that are equipped with homomorphisms $f \colon R \to S$ and $g \colon \mathbb{Z}[X] \to S$, subject to the extra condition that their images centralize one another in $S$ (i.e., if $a = f(r)$ and $b = g(t)$, then $ab = ba$ in $S$).
One way to understand this ring would be to consider its modules/representations.  A right $R[X]$ module $M$ has both a right $R$-action and a $\mathbb{Z}[X]$-action (which we can imagine on either the right or left, as $\mathbb{Z}[X]$ is commutative).  The action of the element $X$ on $M$ commutes with the right $R$-action, so $X$ in fact acts by a right $R$-module endomorphism of $M$.  Conversely, if $M$ is any right $R$-module and $\phi$ is any $R$-module endomorphism of $M$, then we have a unique $R[X]$-module structure with the same $R$-action and with $X$ acting via $\phi$.  So to summarize:

A right $R[X]$-module is a right $R$-module equipped with an $R$-endomorphism.

(Of course, the same sort of statement holds for left $R[X]$-modules.)
I hope this helps you understand the distinction between $R[X]$ and $R * \mathbb{Z}[X]$, as well as the role of $R[X]$.  Let me know if I can clarify anything that I said above.
