Direct sums and Dedekind domains.

Question

Suppose that $D$ is a Dedekind domain and $I,J,K$ are nonzero ideals of $D$. Is there any necessary and sufficient condition for $I \oplus J \cong D \oplus K$?

Answer 1

So the theorem states, if $I_{1},\dots, I_{n}$ and $J_{1},\dots ,J_{m}$ are nonzero fractional ideals of a Dedekind domain $D$ then

$$I_{1}\oplus \cdots \oplus I_{n}\cong J_{1}\oplus \cdots \oplus J_{m}$$ if and only if $n=m$ and $I_{1}\cdots I_{n}=\langle a\rangle J_{1}\cdots J_{m}$, for a nonzero $a$ in the field of fractions of $D$. (The proof can be found in Dummit and Foote, chapter 16.)

So in this case I think we just need $IJ=\langle a\rangle K$ and that should do it.