Question. Is there an example of a ring $R$ (commutative or not) without unity and an element $x \in R$ such that for every $y \in R$ there exists a $z \in R$ such that $y = x z$?
In other words, is there an example of a ring without unity which has an element which divides every other element of the ring on the left? ("On the left" is of course arbitrary.) In a ring with unity, such an element would have a right inverse, and in a commutative ring with unity it would be a unit. What about the non-unital case?
If $R$ and $x \in R$ are an example, then in particular there exists a $z \in R$ such that $x = x z$. I am very unfamiliar with non-unital rings, so I don't know what this implies about $x$.