I would like to know if there a standard term for or well-known applications of the following axiom for rings or semigroups with zero (which is weaker than the "no nontrivial zero divisors" axiom):
if $abc=0$, then $ab=0$ or $bc=0$.
My motivation for looking into this axiom: if i want to view (small) categories as semigroups with $0$, then this is an axiom i need, together with the existence of enough of identities (certain idempotents). In my opinion, from this point of view it is easier to see the analogy between the Yoneda lemma and Cayley's theorem.
If your ring has a unit or if your semigroup is a monoid, then $ac = 0$ implies $a1c= 0$ whence $a1 = 0$ or $1c = 0$, that is $a = 0$ or $c = 0$ and you are back to the traditional definition.
So I am not sure this is really the definition you wish.