identity element is uniqueIdentityElementIsUnique2008-04-24 04:27:272008-07-11 16:07:06Theoremmonoidmonoid% this is the default PlanetMath preamble. as your knowledge
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\textbf{Theorem.}\, The identity element of a monoid is unique.\\
{\em Proof.}\, Let $e$ and $e'$ be identity elements of a monoid \,$(G,\,\cdot)$.\, Since $e$ is an identity element, one has\, $e \cdot e' = e'$.\, Since $e'$ is an identity element, one has also\, $e \cdot e' = e$.\, Thus
$$e' = e \cdot e' = e,$$
i.e. both identity elements are same (in inferring this result from the two first equations, one has used the \PMlinkname{symmetry}{Symmetric} and transitivity of the equality relation).\\
\textbf{Note.}\, The theorem also proves the uniqueness of e.g. the identity element of a group, the \PMlinkname{additive identity}{Ring} 0 of a ring or a field, and the \PMlinkname{multiplicative identity}{Ring} 1 of a field.