Can an element of a noncommutative monoid have exactly one left inverse but no right inverse, or is it like in rings, where this is not possible? If this is possible, what is an example of such a monoid and element?
I don't see why/how the fact about rings or its proof (at least that which I have seen) would generalize to monoids, but I also haven't really worked with monoids yet to know of any concrete example of this.
I appreciate any input, thanks.
Sure, the strings $a^nb^m$ form a monoid under concatenation and the relation $ba=1$.