I am reading Hungerford's Algebra in Chapter 3(Rings), and I am stuck on the following question.
if $S$ is a ring (possibly without identity) with no proper left ideals, then either $S^2=0$ or $S$ is a division ring.
I have found the same question here: if $S$ is a ring (possibly without identity) with no proper left ideals, then either $S^2=0$ or $S$ is a division ring. However, I think his proof provided had some flaws and I cannot fixed it.
Here is what I have reached:
If for any $c$, $d$ in $S$, $cd = 0$, then $S^2 = 0$ obviously holds.
Otherewise, exists $cd \ne 0$, then after some proofs we can get that for all $c$, $d$ in $S$, $cd \ne 0$. Morover, right cancellation in $S$ holds, for any $a \in S$, $b \in S$, $\exists x$ such that $xa = b$. And we can get a right identity as what if $S$ is a ring (possibly without identity) with no proper left ideals, then either $S^2=0$ or $S$ is a division ring. shows. But in the proof of that link, I cannot find a way to get a left cancellation law.
I have some possible approaches to continue: 1. prove right inverse of every element exists; 2. prove the right identity is just the left idnetity.
Can anyone help me with this question? Thanks a lot!