## 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.

 Blog Entries: 6 Recognitions: Gold Member Science Advisor 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.
 Recognitions: Homework Help Science Advisor The canonical example is left and right shift of a sequence or countable dimensional vector space. L(a,b,c,d,...) = (b,c,d,....) R(a,b,c,..)=(0,a,b,c...) LR=id, and RL=/=id. You won't find one in 2x2 matrices - the invertible ones form a group, so there's no point looking.