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

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 $b+P=P$".

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