Rings in which $ab=0$ implies $axb=0$

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 $axb=0$).

The most modern term for this is

$R$ satisfies the $SI$ condition

This is seen in papers by Greg Marks, which I have found to be the most thoughtful and comprehensive recent papers discussing these things. For example , see A taxonomy of 2-primal rings.

Earlier works used the following terms:

$R$ is zero-insertive (zi)

$R$ satisfies the insertion of factors property (IFP)

$R$ is semi-commutative (conflicts with different usages in the literature)

• I am now lost in a maze of twisty ring properties, all alike. Coming from commutative algebra, it fascinates me how people can memorize all these definitions, let alone work with them. Thanks! – Gro-Tsen Apr 27 '16 at 13:06
• Oh, viewing your profile I just learned about the "database of ring properties", which is a great idea! – Gro-Tsen Apr 27 '16 at 13:14
• @Gro-Tsen I'm glad you like it! I really need to get back to work on that website. – rschwieb Apr 27 '16 at 21:21
• @Gro-Tsen well, not many people have these definitions in their heads at all times. This is a great example of a bunch of closely knit conditions that "tease apart" particular aspects of something that happens in a commutative ring. (The nilpotent elements forming an ideal) it's also nice to see the variety of examples it produces – rschwieb Apr 27 '16 at 21:24