Described below are some observations I have made while fiddling around with polynomials. In addition to the two questions below, I am looking for any sort of relevant information so I can read more.
Preliminary question:
Suppose I want to consider the polynomial ring $\mathbb{C}\left[x, y\right]$, except I don't want to assume that $x$ and $y$ are commutative. I believe that the resulting structure is still a ring, but I have never seen anything dealing with such a scenario. Is there a name for such a structure/assumption? Is it even a sound thing to do?
Main question:
Denote by $\mathbb{C}[x, y]^*$ the ring $\mathbb{C}[x, y]$ with the added restriction that $xy \ne yx$ in general. Let $R$ be a non-commutative ring and suppose I have a homomorphism
$$\varphi \colon \mathbb{C} \longrightarrow Z(R)$$
Then it seems I can use some "extended" form of the substitution principle to make the substitution $x = a$ and $y = b$ for any $a, b \in R$, for it appears to be the case that the map
$$\Phi \colon \mathbb{C}[x, y]^* \longrightarrow R$$
given by
$$\Phi(\sum_{i + j = 0}^{n} a_{ij}x^iy^j) = \sum_{i+j=0}^n\varphi(a_{ij})a^ib^j$$
is a homomorphism.
Is this in fact the case or am I missing something?
The usual notation for non-commutative free $k$-algebras is $k\langle x,y\rangle$. And what you have verified is its universal property. All this is well-known, you can find it in Bourbaki's Algèbre for sure. Notice that $k\langle x,y\rangle$ is the tensor algebra over the free module of rank $2$, wheras $k[x,y]$ is the symmetric algebra over the free module of rank $2$.