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) a_ib_i = 1

Let c be an arbitrary element of H. Multiply by 1, and c is the sum of a_ib_ic. Now the elements of H′, when multiplied by anything in H, wind up in R. Each factor cb_i is in R. Therefore c is spanned by the elements a₁a₂a₃ … a_n. since c was arbitrary, H is finitely generated.

Dedekind Implies Noetherian

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.