One-sided zero-divisors which are identity from the other side

Question

What are examples (if any) of a ring $R$ with elements $x,y \in R$ such that $xy=0$ yet $yx=1$?

Is there a characterization of rings which exhibit such pairs of elements? Do such pairs have a special name?

Answer 1

In any associative unital ring, these conditions imply that $x=0$, since $$0=0x=xyx=x1=x;$$ and then $1=yx=y0=0$. So this can only hold in the trivial ring.

Answer 2

Assuming associativity and two-sided $1$ and $0$, there is only one such ring, because$$1=1\cdot1\\=(yx)(yx)\\=y(xy)x\\=x\cdot 0\cdot y=0$$