# For a monoid, if uv = 1, do we know vu = 1?

If $M$ is a monoid and $u,v\in M$, and $uv = 1$, do we know $vu = 1$? Can someone prove this or provide a counterexample? I tried to come up with one (a counterexample, that is) using 2 x 2 matrices but was unsuccessful.

jambaugh
Gold Member
Assume:
$$uv=1$$
If there exists a w such that:
$$vw=1$$
Then by associativity:
$$w=1w=(uv)w=u(vw)=u1=u$$
Thus if such a w exists it must be u.

This isn't quite enough but I can't find a short proof either. I'll think on it.

matt grime