For a prime ring $R$, you can define its "*Martindale ring of quotients*" $Q(R)$. See for example:

*Martindale, Wallace S. III*, **Prime rings satisfying a generalized polynomial identity**, J. Algebra 12, 576-584 (1969). ZBL0175.03102.

The center of this ring is called the "*extended centroid*" of $R$. It is known that it is a field.

I was wondering if anyone knows what is the extended centroid of $R = K\left<x_1,x_2,\dots,x_n\right>$, the free associative algebra over a field $K$? Is it just $K$ in this case?

In the literature (as the title of the article above suggests), this construction is usually considered in the case of rings with a polynomial identity. I cannot find any mention of the construction for free algebras.