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The ring of matrix is not an integral domain. How to prove that the inverse is unique?

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You don't need it to be an integral domain to have unique inverses. –  mez May 22 at 18:34
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up vote 17 down vote accepted

If $ab=ba=1$, and if also $ac=ca=1$, you have $c=(ba)c=b(ac)=b$. In fact, if you reread it, this shows that whenever you know both a left inverse exists and a right inverse exists, then actually they are the same element, so it is a two-sided inverse and it is unique.

It has little to do with integral domains: it's true for any associative operation with an identity.

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You could also mention that your argument actually shows that if a left inverse and a right inverse exist, then they are equal and so he inverse exists and is unique. –  Martin Argerami May 22 at 18:52
    
@MartinArgerami Good idea: I added that. Thanks! –  rschwieb May 22 at 19:17
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