K-Algebras in which every subalgebra generated by single element is one-dimensional

Question

I am looking for informations (structural theorems, classifications, examples) about K-algebras (K is a field of arbitrary characteristic) in which the following condition holds: $$\forall_{x\in A}\exists_{k\in K} x^2=kx.$$ I do not know if such algebras have a common name in literature. An obvious example here is $K=\mathbb{Q}$ and $A=\mathbb{Q}$ and all subalgebras of $A$ (for instance $\mathbb{Z}$). Does there exists a noncommutative algebra of this kind? What if we assume that $A$ is a domain? (integral domain? reduced?). I will be grateful for information on this topic (links to articles, lecture notes). Unfortunately, I could not find anything valuable. Thank you in advance