Localization by a prime ideal

Question

Let $A$ be a commutative ring with identy and let $P\subseteq A$ a prime ideal. We consider $$A_P:=\bigg\{\frac{a}{s}\:\bigg|\; a\in A, s\in A\setminus P\bigg\}.$$

I must prove that $$\frac{a}{s}\;\text{is a unit in}\; A_P\iff a\in A\setminus P$$

$(\Rightarrow)$ If $\frac{a}{b}\in A_P$ is a unit, exists $\frac{b}{t}\in A_P$ such that $$\frac{a}{s}\frac{b}{t}=1\implies (ab,st)\equiv (1,1),$$ therefore exists $u\in A\setminus P$ such that $u(ab-st)=0$, then $$a(ub)=ust\in A\setminus P$$, since $P$ is prime $a\in A\setminus P$.

for the other implication?

could someone suggest me something? I tried, but I can't.

Thanks!

Answer 1

The other direction is even easier: the point is that we can write $a$ into the denominator, and we have $\dfrac sa\dfrac as=1$.