Let $A$ be a commutative ring and $S$ a submonoid of the multiplicative structure of $A$. Then consider the quotient ring of $A$ by $S$, denoted $S^{-1}A$, see e.g. Lang p.107. Is it true that there is a bijection between the ideals of $A$ and the ideals of $S^{-1}A$? I have proved so, but i want to verify it.
Thanks.
Added(1): Consider the map that takes an ideal $\alpha$ of $A$ to the ideal of $S^{-1}A$ denoted $S^{-1} \alpha = \left\{\frac{x}{s} : x \in \alpha, s \in S\right\}$. Is this map surjective?
Added(2): Is the above mentioned map injective only if restricted on ideals that are prime and do not intersect with $S$?