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 ?



Existence of a generalized matrix inverse over an arbitrary field?
The condition $AA'A=A$ means that the semigroup of matrices is regular. This is really true: see Clifford - Preston, The Algebraic Theory of Semigroups, sec.2.2, ex. 6(g).


