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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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marked as duplicate by Arturo Magidin, Bill Cook, anon, Srivatsan, Zev Chonoles Jan 30 '12 at 4:45

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

No. See this previous question. Though it is about rings, forgetting the addition will give you the examples for monoids. –  Arturo Magidin Jan 30 '12 at 1:00

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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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You've only demonstrated that $a$ has a right inverse and that $b$ has a left inverse. Could you please show why $(bk)a=1$ and $b(ka)=1$ as well? Before you try, be sure to consider the example here. –  Arturo Magidin Jan 30 '12 at 1:11
@ArturoMagidin Ah, I just saw your comment and was reading through it. Thank you. I will edit accordingly! –  Galois Group Jan 30 '12 at 1:13
@ArturoMagidin If we know av=bv implies a=b, then this converse holds. v(kuv)=v implies (vku)v=v, which means vku=1, that is v(ku)=1. –  Shannon Jan 30 '12 at 2:02
@Shannon: Sure, extra hypothesis may suffice, but it's not true in general. –  Arturo Magidin Jan 30 '12 at 2:35

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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Is not unit a standard term for invertible element? –  Galois Group Aug 5 '12 at 6:30
@FortuonPaendrag: Yes, in a ring. But it must have surprised me to see the terminology used in a non-ring setting. –  Henning Makholm Aug 5 '12 at 12:28

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