Dedekind Domains, Invertible Ideals are Finitely Generated
Invertible Ideals are Finitely Generated
Let H be an invertible ideal with inverse H′.
We know that H times H′ generates 1.
Write the following equation, with generators a and b drawn from H and H′ respectively.
sum(i=1 to n) aibi = 1
Let c be an arbitrary element of H.
Multiply by 1, and c is the sum of aibic.
Now the elements of H′, when multiplied by anything in H, wind up in R.
Each factor cbi is in R.
Therefore c is spanned by the elements a1a2a3 … an.
since c was arbitrary, H is finitely generated.
In the last section we showed every ideal in a dedekind domain R is invertible.
Now we know all invertible ideals are finitely generated.
Therefore all the ideals of R are finitely generated, and R is noetherian.