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.