In this question, a ring is not supposed to be commutative.
Let $K=\mathbb{Q}[i]$, and $A$ a subalgebra of $\mathcal{M}_n(K)$ (the algebra of $n \times n$ matrices). We suppose $A$ is a division ring. Is $A$ necessarily commutative ?
More generally, if $K$ is an algebraic extension of $\mathbb{Q}$, where $-1$ is a square, is $A$ necessarily commutative ?
If $-1$ is not the sum of three square in $K$, we can construct a division ring, that is an algebraic extension of $K$. The construction is similar to the construction of quaternions over $\mathbb{R}$. So, $A$ is not necessarily commutative in this case.
Thanks in advance.