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If $AR=I$ ($R$ is the right inverse) and $LA=I$ (L is the left inverse), how can we show that $L=R$? I am a bit skeptic about the statement but stuck at the moment in terms of showing that $L=R$. Any help would be great!

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Hint: $LAR=\,?$

$L = LI = L(AR) = (LA)R = IR = R$

It works in any monoid (closed under operation, operation is associative, operation has identity element): If an element has a right inverse and a left inverse, then they are equal.

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