Here is the proposition I want to prove:
There is a one-to-one order preserving correspondence between the ideals $\mathfrak{b}$ of $A$ which contain $\mathfrak{a},$ and the ideals $\bar{\mathfrak{b}}$ of $A/\mathfrak{a},$ given by $\phi^{-1}(\bar{\mathfrak{b}}) = {\mathfrak{b}}.$
But, I am not quite sure:
1- How can I prove the "one-to-one order preserving correspondence", what does this statement means? Should I find a bijection between $\mathfrak{b}$ and $\bar{\mathfrak{b}}$? how can I do this?
2- What does it mean "order preserving"?
Could someone help me please in this?
Hint Use the "canonical" projection $\rho:A\twoheadrightarrow A/\mathfrak a$, which sends each $x\in A$ to the coset $x+\mathfrak a$ in $A/\mathfrak a$.
The bijection is between the sets $\{\mathfrak b:\mathfrak a\subset \mathfrak b\subset A \,\text {is an ideal}\}$ and $\{\bar{\mathfrak b}:\bar {\mathfrak b}\subset A/\mathfrak a\ \text{is an ideal}\}$.