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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?

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twisted polynomial or noncommutative polynomial ring. whether it is sound is a matter of taste, but it can be done at any rate –  Alexander Grothendieck Nov 17 '13 at 22:27
    
For a field $F$ and indeterminates $x,y$, the free $F$-algebra on $x,y$ is denoted $F\langle x,y \rangle$. In a sense it is "the most general (associative) $F$-algebra containing $x$ and $y$." The ordinary polynomial ring $F[x,y]$ can be retrieved as a quotient of the free algebra: $F[x,y]=F\langle x,y \rangle/(xy-yx)$. –  rschwieb Nov 18 '13 at 14:29
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1 Answer 1

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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 $2$.

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