# Generalizing polynomial identities for rings

For a ring $$R$$, a polynomial identity of $$R$$ is a polynomial (in non-commuting variables) $$f(x_1,\ldots,x_n)\in \mathbb{Z}[x_1,\ldots, x_n]$$ such that for any choice of $$a_i\in R$$, $$f(a_1,\ldots, a_n)=0$$.

For example, for all $$n$$, the standard (or symmetric) polynomial $$S_{2n}(x_1,\ldots,x_{2n})=\sum_{\sigma\in S_{2n}}$$sign$$(\sigma)x_{\sigma(1)}\cdots x_{\sigma(2n)}$$ is a polynomial identity for the ring $$R=M_n(\mathbb{C})$$.

I would like to understand how this idea might extend to considering, rather than polynomials in the $$x_i$$, polynomials in the $$x_i$$ and their inverses, i.e., an element of $$\mathbb{Z}\langle x_1,\ldots,x_n\rangle$$.

Obviously, to even be defined, one would need to restrict any 'generalized' polynomial identity to the units of $$R$$. An easy example is:

Example: If $$R^\times$$ is abelian, then $$f(x,y) = 1 - xyx^{-1}y^{-1}$$ is such a generalized polynomial identity. Similar statements can be made if $$R^\times$$ is nilpotent, solvable, &c.

I'm primarily interested in the situation of $$R=M_n(\mathbb{C})$$, with $$R^\times=\textrm{GL}_n(\mathbb{C})$$. Is anyone aware of previous work on this subject? Are there any 'obvious' such identities?

More optimistically, I wonder if anything might be said by taking the (skew) field of fractions of $$\mathbb{Z}[x_1,\ldots,x_n]$$ and asking what identities arise here. For example, something of the form $$f(x,y)=x(x-y)^{-1}y$$. In this case, the domain of the identity $$f$$ would need to be restricted to those $$x,y\in R$$ so that $$x-y\in R^\times$$. More generally, the domain might become even stranger (consider $$f(x)=(1+(1+x)^{-1})^{-1}$$, $$f(x)=(1+(1+(1+x)^{-1})^{-1})^{-1}$$, ...).

I'll add that I'm not particularly worried about how nasty this domain of definition may be for a particular identity, just whether such an identity exists with nonempty domain; again, I'm interested in $$R=\textrm{GL}_n(\mathbb{C})$$, and here (I believe) for any fixed such $$f$$, the domain will be nonempty unless some inverted term is itself an identity.

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