Let $C$ be a cancellative semigroup (that is a semigroup in which, for all $a,b c, \, ca = cb \; \Rightarrow \; a = b$ and $ac = bc \; \Rightarrow \; a = b$) and suppose that $C$ has no identity. Then there is no pair of element $e, a$ in $C$ for which $ea = a$ or for which $ae = a$.
I want to show by contradiction. If there exist $a, e $ in $C$ such that $ea = a$. My aim to show that $e$ is the identity element of $C$. How to proceed further.
Any help would be appreciated. Thank you.
Suppose there exist $e$ and $a$ such that $ea = a$. Then $eea = ea = a$, whence $ee = e$ by right cancellation. Let now $x$ be any element. Then $eex = ex$ and $xee = xe$, whence by left (respectively right) cancellation, $ex = x$ and $xe = x$. Thus $e$ is an identity, a contradiction. Dual proof if there exist $e$ and $a$ such that $ae = a$