Some questions from a proof involves integral ideal

Question

In this proof, I don't understand

(1) Why by multiplying $a'$ by a principal ideal we may suppose $a={a'}^{-1}$ is an integral ideal?

(2) Why is $b$ is an integral ideal in the same class as $a'$? For (1) I guess we just multiply $a'$ by a principal ideal generated by its common denominator then $a'$ can be integral, but does that imply the inverse of the internal ideal is integral? For (2), I have no idea.

Answer 1

(1) Any fractional ideal of $ K $ is a finitely generated $ \mathcal O_K $-submodule of $ K $. Since $ K $ is the fraction field of $ \mathcal O_K $, you may simply clear denominators of the finitely many generators of a fractional ideal by multiplying by an element of $ \mathcal O_K $.

(2) $ \mathfrak a^{-1} = \mathfrak a' $, and $ x $ is an element of $ K $.