I'm sure there must be some standard term for (not necessarily commutative) rings $R$ in which $ab=0$ implies $(\forall x)\, axb=0$ (for example, this is the case if $R$ is commutative or is a domain). What is this term?
Additionally, or alternatively, what about (two-sided) ideals $I$ such that $ab\in I$ implies $(\forall x)\, axb\in I$, i.e., ideals quotienting by which gives a ring as I just said? Do they have a name?
Edit: I should probably also mention the stronger condition that $ab=0$ implies $ba=0$: such rings are called "reversible" (Cohn, "Reversible Rings", Bull. London Math. Soc. 31 (1999), 641–648). Clearly, commutative rings and domains are reversible, and reversible rings satisfy the property I'm looking for a name for (because in a reversible ring, if $ab=0$ then $ba=0$ so $bax=0$ so $axb=0$).