# What is the extended centroid of a free algebra?

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 $$R = K\left$$, 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.

that the extended centroid of $$K\langle x_1,\dots,x_n\rangle$$ is just $$K$$, as you speculated.