A question on infinite local rings which are not division ring

Question

Is it true that if $(R,m)$ is a (not necessary commutative) local ring then $R$ and $m$ have the same cardinal ? (Exclude TWO trivial cases: when $R$ is finite and when $R$ is a division ring)

On the other hand if the ring is Artinian then the above claim is true.

Answer 1

If $m\neq0$ then pick $0\neq x\in m$. The map $r\mapsto xr$ is a nonzero right module homomorphism from $R$ to $m$. So $R/m$ is a subquotient (as right module) of $m$ and so $|m|\geq |R/m|$.

But $|R|=|m||R/m|$, so if $R$ is infinite then $|m|=|R|$.