System of homogen linear equations in a division ring Let $K$ be a division ring (one does not suppose that $K$ is commutative) and $m,n$ two positive integers such that $m<n$. Consider the system of homogen linear equations $$\left\\{\begin{array}{rcl} a_{11}x_1+a_{12}x_2+\cdots+a_{1n}x_n&=&0\\\ a_{21}x_1+a_{22}x_2+\cdots+a_{2n}x_n&=&0\\\ \vdots\qquad\qquad\qquad&\vdots&\vdots\\\ a_{m1}x_1+a_{m2}x_2+\cdots+a_{mn}x_n&=&0\\\ \end{array}\right.$$ where the $a_{ij}$'s are in $K$. Can we assert that this system admits an infinity of solutions. Obviously, it is true when $K$ is commutative. But when $K$ is not, I do not know if this results still holds. Is it also true in the non commutative case? Thanks in advance.

The system has infinitely many solutions, even when $K$ is non-commutative. The reason is that row reduction is still possible over a non-commutative division ring.

* * *

Side remark: if $K$ is commutative, the system has infinitely many solutions only if $K$ is infinite. If $K$ is not commutative, then it is automatically infinite by Wedderburn's little theorem.