Let $D$ be a division ring with the center $F$. Suppose that $G$ is a subgroup of the multiplicative group of $D$ such that every element of $G$ is algebraic over $F$. Then may we conclude that any element of $F[G]$ is also algebraic over $F$ ?

Here $F[G]$ stands for the subring of $D$ generated by $G \cup F$. An element of $F[G]$ may be written in the form $\sum\limits_{i = 1}^n {{a_i}{g_i}} $, where $a_i \in F$ and $g_i \in G$ for some integer $n$.

We see that if $D$ is a (commutative) field, then from Field theory, the answer is 'yes'. In the case $D$ is not commutative, I do not know the answer.

Thanks for any replies!


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