A ring $R$ is said to be left unital if there exists an element $e\in R$ such that $ex=x\forall x\in R$.
Show that if for some $a\in R$ we have $aR=Ra=R$ then $R$ is left-unital.
I need to show existence of some $e\in R$.
Let $x\in R=Ra\implies x=ra$. But I am not getting how to proceed from here.
Since $Ra = R$, there exists $r \in R$ such that $ra = a$. Therefore $r$ is a candidate as a left-unit. It remains to see that $rb = b$ for every element $b \in R$.
Indeed, if $b \in R$, since $aR = R$, there exists $q \in R$ such that $b = aq = raq = rb$, yielding $R$ is left-unital with $e = r$.