Problem: For a general ring $R$ I want to show that: if $R$ has a unique right unity, then it has an overall unity.
Attempt: Suppose $e$ is the unique right unity of $R$. Then for any $r \in R$, we have that $re=r$ and so $er=ere$ and $er-ere=0$. From here I feel the answer should be obvious but I'm not seeing it right away. Help appreciated.
The key here is that it is unique. Suppose there is no element $s$ such that $sr=rs=r$ for all $r$. Then there exists $r$ such that $er\neq r$. Then $e'=e+er-r$ is a distinct right identity, which is a contradiction.