Prime Ideals and multiplicative sets I am currently studying a course on commutative algebra and came across this statement: 

An Ideal $I$ in a ring $R$ is prime if and only if $R\setminus I$ is a multiplicative set.

I have proved it the following way and was wondering if anyone would be kind enough to go through the proof to check for any errors.
So assume $R\setminus I$ is multiplicative, then $f\in R\setminus I$, $g \in R\setminus I\implies fg \in R\setminus I$. This is the same as saying $f,g$ not in $I$ implies $f.g$ not in $I$. So we just take the contrapositive of this to get our implication that $f.g$ $\in$ $I$ $\implies$ $f$ $\in$ $I$ or $g$ $\in$ $I$. Hence I is prime.
Now suppose $I$ is prime. Then,
$f.g$ $\in$ $I$ $\implies$ $f$ $\in$ $I$ or $g$ $\in$ $I$. Again taking the contrapositive of this, we can deduce the relation from above which would show that $R\setminus I$ is indeed multiplicative.
Is this a valid enough proof or would I need to be more "rigourous"?
 A: I correct one direction. can you correct the other?  

I use Transposition: $P \implies Q  \equiv  (\lnot Q\implies \lnot P)$. (and De Morgan's Theorem)  

Assume $R\setminus I$ is multiplicative set. So:
 $$(f \in   R\setminus I) \land  (g\in R\setminus I)\implies f g \in R\setminus I.$$ This is the same as saying
$$f g \notin R\setminus I\implies (f\notin R\setminus I) \lor (g\notin R\setminus I)$$ This is the same as saying $$fg\in I\implies (f\in I) \lor (g\in I).$$ So $I$ is is prime.
A: Multiplicativity of $\bar P\,$ is just the contrapositive of primality of $P$
$\begin{align}\ \ P\,\ {\rm prime} \iff&\ ab\in P\Rightarrow a\in P\,\ {\rm or}\,\  b\in P\\ \iff&\ a,b\in \bar P\Rightarrow\,ab\in \bar P\quad \rm [contrapositive\ of\ prior]\\ \iff&\ \bar P\,\ {\rm is\ closed\ under\ multiplication}\end{align}$
Note  $\bar P$ closed under multiplication is, in $R/P,\,$ $\,a,b\neq 0\,\Rightarrow\,ab\neq 0,\,$ i.e. the domain property.
Remark $\ $ More generally: $S$ is multiplicative and saturated (closed under divisors) $\iff$ its complement is a union of prime ideals. This becomes clearer when one studies localizations
