There is a well known theorem saying that a commutative field that is finitely generated as a ring has to be finite (Kaplansky).

Is the same true for non-commutative "fields" (usually called skewfields or division rings)?

I have searched for this problem on the internet, but the only reference I have found was a notice on a related question in the book of P.M.Cohn: Skew Fields - Theory of General Division Rings (1995) p. 412 here:

Let $L$ be a skewfield finitely generated as a ring over a subfield $K$. Is $L$ necessarily algebraic over $K$?

At that time this question was open as well. Does anybody know some recent references on any of these problems? Thank you in advance.


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