Let $A\in M_n(K)$ be a square matrix over a field $K$. The notion of inverse matrix was generalized by Moore and Penrose for real and complex matrices (also called pseudo-inverse $A^{\dagger}$ of $A$, satisfying $AA^{\dagger}A=A, A^{\dagger}AA^{\dagger}=A^{\dagger}$ and $AA^{\dagger}$ Hermitian). This was again generalized to arbitrary fields with involutory automorphism of $K$ by Pearl, but the existence depends on additional rank conditions.

**Question:** What is known on the existence of a *generalized inverse* $A'$, which just satisfies $AA'A=A$, over an arbitrary field ?
Do we also need an additional assumption, perhaps on the characteristic of $K$, or again some rank condition ?