Let $\mathbb{k}$ be a field. For the purposes of this question, a division ring is a finite-dimensional $\mathbb{k}$-algebra $A$ in which every non-zero object is invertible.
Is there a commonly accepted name for such objects?
What are some examples? More precisely, are there other examples than the ones below?
Clearly, if $\mathbb{k}$ is algebraically closed, any division ring is isomorphic to $\mathbb{k}$. Hence, these fields satisfy the desired property.
Another more interesting example is given by $\mathbb{R}$. Namely, it is known that every division ring over $\mathbb{R}$ is isomorphic to $\mathbb{R}$, $\mathbb{C}$, or $\mathbb{H}$.