# Terminology for a ring where every right cancellable element is cancellable

Is there any standard terminology for a ring in which every right cancellable element is cancellable (or equivalently, every left zero divisor is a zero divisor)? I'm aware of some people going for the term "left eversible", but the term seems not so well established (and I'm not particularly fond of it).

It would be natural to call a monoid $$H$$ with the same property hopfian: This would be consistent with the abstract definition of a hopfian object in a category [Wiki.en], when regarding $$H$$ as a one-object category in the usual fashion (or maybe I'm swapping left for right, I guess much depends on the convention adopted for the order of composition of two morphisms). But a hopfian ring isn't usually a ring whose multiplicative monoid is hopfian; and on the other hand, the term "hopfian monoid" is already taken for something else (see Benjamin Steinberg's comments).

• Hopfian monoid has a different well established meaning so please don't use it. With your definition all groups are Hopfian. A monoid is Hopfian if every surjective monoid homomorphism is injective. Jun 2 at 16:25
• Left reversible also has another meaning in semigroup theory that is well established. Jun 2 at 16:25
• Fine, I'll not use "hopfian" for my purposes. As for your 2nd comment, I really meant "eversible" (not "reversible"). Jun 2 at 16:58

I don't think that there is a "standard terminology" for a ring $$R$$ whose left zero divisors are also right zero divisors. The "reversible rings" that P.M.Cohn considers here are rings satisfying the stronger condition $$ab=0 \Rightarrow ba=0$$. Usually in the theory of Ore localization, you would call a multiplicative subset $$S$$ of a ring $$R$$, "left reversible" if $$rs=0 \Rightarrow \exists s'\in S: s'r=0$$ for any $$r\in R$$ and $$s\in S$$ (see for instance here). Hence what you want is that $$S=R\setminus\{0\}$$ is a left reversible set in the sense of Ore localization, but this would somehow clash with Cohn's terminology.