We know that for a commutative ring $R$ with identity $1_R$,
(a) An ideal $P$ is prime iff $R/P$ is an integral domain
(b) An ideal $M$ is maximal iff $R/M$ is a field.
The way i proved the statement (a) doesn't need the existence of $1_R$. But then the way i proved (b) uses the existence of $1_R$. Also for the proof of "Every maximal ideal is prime" I used (a),(b)
(Note that if we take $R=x\mathbb{R}[x],I=x^2\mathbb{R}[x]$ then $I$ is a maximal ideal of $R$ which is not prime. Indeed, assume $I\subsetneq M$ then we can show $\exists ax\in M\text{ for some }a\in\mathbb{R}^*$. Thus $M=R$. But $x.x=x^2\in I$ but $x\not\in I$ which shows that $I$ is not prime). So I used the existence of $1_R$.
When I proved the existence of "Maximal ideal", i used Zorn's lemma along with the existence of $1_R$.
My question: I can formulate the characterization of prime ideal in an arbitrary ring as
"An ideal $P$ is prime iff $R/P$ has no zero-divisor."
(I hope it is correct! If not, please tell me where i am wrong.)
So, the question is how to characterize the maximal ideal of a ring $R$ if
(i) $R$ is commutative but has no unity (For this part I assume that a maximal ideal $M$ is given. As in an arbitrary ring maximal ideals may not exist, I'm assuming the existence and want to conclude about the ring $R/M$.)
(ii) $R$ is not commutative but has unity (I think $R/P$ is division ring, is necessary and sufficient. If I'm wrong, please correct my statement)
(iii) $R$ is non-commutative and without unity.