Given a rng $(R,+,\cdot)$ and $e \in R$ with $\forall a \in R: e \cdot a = a$, I would like to prove $a \cdot e = a$. What I have so far is the following:
Assume there exists $r \in R$ with $\forall a \in R: a \cdot r = a$. Plug in $e$ for $a$ to get $e \cdot r = e$. But by the given conditions, $e \cdot r = r$. Thus, $r=e$ and consequently $e \cdot r = a$.
Now the problem I have with this is that I have to assume there exists a right identity element. Any tips on how I could solve this problem? Is the thing I'm trying to prove even true? I made this exercise for myself, but my feeling tells me it's true.