Suppose that $R$ is a ring, not necessarily commutative nor associative. Assume that for every non-zero $a \in R$, the left multiplication map
$$ \lambda_a \colon R \to R \colon x \mapsto ax $$
is invertible. (We do *not* assume that its inverse is again a left multiplication map $\lambda_b$ for some $b \in R$.)

Is such a ring $R$ necessarily unital (i.e., does it have a unit $1 \in R$), and is it then a division ring (i.e., does every element $a$ have a two-sided inverse)?

Note: *if* $R$ is unital and associative and every $\lambda_a$ ($a \neq 0$) is invertible, then it is indeed a division ring, and the inverse of $\lambda_a$ is equal to $\lambda_b$ where $b=a^{-1}$.