I'm having trouble with this homework problem (from Algebra by Hungerford).
If $S$ is a ring (possibly without identity) with no proper left ideals, then either $S^2=0$ or $S$ is a division ring.
We've just proven in the previous part that if $S$ is a ring with identity and with no proper left ideals, then $S$ is a division ring. So the strategy is to suppose $S^2 \neq 0$ and prove that $S$ has an identity. The problem also has a hint:
Hint: Show that $\{a \in S \mid Sa = 0\}$ is an ideal. If $cd \neq 0$, show that $\{r \in S \mid rd = 0\} = 0$. Find $e \in S$ such that $ed = d$ and show that $e$ is a two-sided identity.
Here's what I have so far, with some boring details removed:
Note that $S_0 = \{a \in S \mid Sa = 0\}$ is an ideal, because [proof it's an ideal].
Since $S$ has no proper left ideals, we must have $S_0 = 0$ or $S_0 = S$. This means that either $S^2 = 0$ (if $S_0 = S$) or for every non-zero $a \in S$, there is some $b \in S$ with $ba \neq 0$ (if $S_0 = 0$). From now on suppose $S_0 = 0$, and we will show that $S$ is a division ring.
Note also that $A(d) = \{r \in S \mid rd = 0\}$ is a left ideal, because [proof it's a left ideal].
Now if $cd \neq 0$ for some $c, d \in S$, then $A(d) \neq S$ because $c \notin A(d)$; but $S$ has no proper left ideals, so $A(d) = 0$. Then $rd = sd$ if and only if $r = s$, for otherwise $(r-s)d = 0$ contradicts $A(d) = 0$.
Consider the left ideal $I_a = \{ra \mid r \in S\}$. [proof that it's a left ideal]. Since $S$ has no proper left ideals, we must have $I_a = 0$ or $I_a = S$ for all $a \in S$.
If $cd \neq 0$ then $I_d = S$ (so $I_d = S$ for all non-zero $d \in S$, since we showed that for all $d$ there is a $c$ with $cd \neq 0$), and therefore $d \in I_d$ so there is some $e \in S$ with $ed = d$. As we observed before, $rd = sd$ iff $r = s$, and $(re)d = r(ed) = rd$, so $re = r$ for each $r \in S$.
In summary, for every $a \in S$ we have some $e \in S$ (specific to this $a$) such that $e a = a$ and $r e = r$ for all $r \in S$. I feel like it should be very easy to show that there is a two-sided identity (e.g. by showing that all of these $e$ are in fact the same), but I can't seem to get there.
Thanks for any help!