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If $xy$ is a unit, are $x$ and $y$ units?

There is no doubt if a and b are units then ab is a unit. How about the converse? Still holds?

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    No. See [this previous question](http://math.stackexchange.com/questions/99949/if-xy-is-a-unit-are-x-and-y-units/99951#99951). Though it is about rings, forgetting the addition will give you the examples for monoids.2012-01-30

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This is untrue in general, but $a$ and $b$ do have one-sided inverses.

Proof: If for $a,b \in M$, $ab$ is a unit, then $\exists k \in M$ such that $(ab)k=k(ab)=1$.Since multiplication in a momoid is associative, we have that $a(bk)=1$ and $(ka)b=1$, demonstrating explicitly a right inverse for $a$ and a left inverse for $b$. But as Arturo Magidin's example-which he has linked to-demonstrates, this may be the best we can obtain.

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    @Shannon: Sure, extra hypothesis may suffice, but it's not true in general.2012-01-30
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What do you mean by "unit"? Usually it means the identity element, but there can only be one of those in a monoid, so I'll assume you mean an invertible element.

If the monoid is commutative, then $ab$ invertible easily implies $a$ invertible and $b$ invertible.

In a non-commutative monoid: if $ab$ is right invertible, then $a$ is also right invertible, but we cannot say anything about $b$ in particular. (For example, in the monoid generated by $a$, $b$ and $c$ with the single relation $abc=1$, $ab$ is right invertible, but $b$ is neither left nor right invertible).

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    @FortuonPaendrag: Yes, in a ring. But it must have surprised me to see the terminology used in a non-ring setting.2012-08-05