Polynomials in the noncommutative case

Let $R$ be a non-commutative ring. Then we can define the semigroup ring of $R$ over the free non-commutative semigroup $F^{\rightarrow}(A)$. Then this semigroup ring $RF^{\rightarrow}(A)$ consists of ordered words over the alphabet $A$ with coefficients in $R$. This mirrors the construction of the polynomial ring $R[x_1,x_2,...]$ in commutative algebra as a free commutative semigroup ring.

Elements of the free non-commutative semigroup ring $RF^{\rightarrow}(x,y,z)$ consist of ordered words with coefficients in $R$. For example, we might get a polynomial like $5xyx + 6xz + 7zx$. A term like $xyx$ is not the same as $x^2y$ and $xz$ and $zx$ are not the same as one another. This non-commutativity would naturally add a great deal of complexity to compuations over non-commutative rings.

The issue with this free semigroup ring, is that it is not truly the ring of polynomial functions over $R$. Consider $c_1,c_2,...$ to be constants and $x,y,z,...$ to be indeterminates. Then a coefficient may not commute with its indeterminate, and $c_1 x \not= xc_1$. So for polynomial functions over a non-commutative ring you ultimately get potentially nasty terms like $c_1 x c_2 c_3 y c_4 x c_5 z c_6$.

A term of a polynomial function in a non-commutative ring is an alternating sequence of coefficients and indeterminates, where the coefficients can be identities. So for example $c_1xyc_2yxc_3$ is a term except the coefficients between $x$ and $y$ and then between $y$ and $x$ are simply identities, but there is always the possibility of having coefficients between indeterminates.

The terms of the free semigroup ring are already complicated enough, but terms in the ring of polynomial functions with scattered coefficients are even more messy. This is a serious impediment to doing noncommutative algebraic geometry, so what is the big idea? It seems that noncommutative geometry has more to do with functional analysis and operator theory then polynomials.

Noncommutative spaces can instead be studied using concepts of functional analysis and operator theory like C*-algebras. In particular, the Gelfand representation related C* algebras to locally compact Hausdorff spaces. The study of noncommutative geometry based upon techniques from functional analysis is an interesting direction, albiet one that is very different from you might expect.