I have problem understanding the proof of the statement in Lang's algebraic number theory.
The proof explicitly finds the generator $\alpha $ of given ideal $\mathfrak{a} = \mathfrak{p}_1^{e_1} \dots \mathfrak{p}_s^{e_s}$ in the Dedekind ring $A $. ($\mathfrak{p}_i$'s being the primes in A)
Let $\pi _i$ be the elements such that it is in $\mathfrak{p}_i$ but not in $\mathfrak{p}_i ^2 $. Using Chinese Remainder Theorem, find $\alpha$ such that $\alpha \equiv \pi_i^{e_i} \mod{\mathfrak{p_i^{e_i+1}}} $. The book then says that the $\alpha$ is the generator for the ideal, but I don't get why $\alpha $ is the generator.