I'm trying to prove => part of atiyah 9.8.
I understood everything but 
Can somebody help me to understand why does it suffice to show for integral ideal of Ap?
2026-03-27 08:46:05.1774601165
Theorem 9.8 of Atiyah : why does it suffice to show for integral ideal in Ap?
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I know why it suffices to show integral ideal (instead of fractional ideal).
Suppose every integral ideal is invertible - (1).
Let M be a nonzero fractional ideal of A. Then there is nonzero x in A s.t. xM is in A. Now since xM is integral ideal, there is a A-submodule N of Frac(A) s.t. (xM)N=A by (1). Note that xN is a A-submodule N of Frac(A) and M(xN)=(xM)N=A.
So we can show that every fractional ideal is invertible.