Verifying proof :an Ideal $P$ is prime Ideal if $R/P$ is an integral domain.

Question

I had to write the proof to show that an Ideal $P$ of a commutative ring $R$ is prime Ideal if $R/P$ is an integral domain.

let $a,b\in R$ s.t. $ab\in P$ ,
$$ab+P=P\implies(a+P)(b+P)=P\implies\overline a \overline b=0$$ and as $R/P$ is an integral domain either $a=0$ or $b=0$ $\implies a+P=P$ or $b+P=P$.

$\therefore$ either $a\in P$ or $b\in P$.
Hence, $P$ is a prime Ideal.

Am I correct in writing this or am I making some error...

Answer 1

Looks good. Probably worth doing both directions; this is an if-and-only-if statement.

Answer 2

Your proof is mathematically correct, but one suggestion I would make is to make sure to define all of the notation that you use. For example, I assume that when you write $\overline{a}$, you mean the equivalence class of $a$ in the quotient ring $R/P$. It might be good to state that in your proof. Another suggestion would be to avoid using the $\Rightarrow$ symbol when you are writing a sentence in English. Instead of writing "as $R/P$ is an integral domain either $a=0$ or $b=0 \Rightarrow a+P=P$ or $b+P=P$", I would suggest writing "as $R/P$ is an integral domain, either $a=0$ or $b=0$, which implies that either $a+P=P$ or $b+P=P$".