  We introduce a weak division-like property for noncommutative rings: a
nontrivial ring is fadelian if for all nonzero $a,x$ there exist $b,c$ such
that $x=ab+ca$. We prove properties of fadelian rings, and construct examples
of such rings which are not division rings, as well as non-Noetherian and
non-Ore examples. We have also formalized some of these results in the Lean
proof assistant.
