Properties of the first syzygy over Dedekind domains

Question

Let $L$ be a Dedekind domain and $I = (a) + (b)$ a non-principal proper ideal in $L$. Consider equation of the form $xa + yb = 0$. Is it true that then there exists a proper ideal $J$ such that any $x, y$ that satisfy the equation belong in $J$?

Answer 1

Let $M\subset L$ be any maximal ideal. We will show that there exists $x,y\in L$ with $xa+yb=0$ and $(x,y)$ is not contained in $M$, which will show that no such proper ideal $J$ as in your question exists.

If you localize at $M$, then $L_M$ is a pid and thus $(a,b)$ is principal, generated by either $a$ or $b$. Wlog, assume $a$ generates it and then $b=pa$ for some $p\in L_M$, which we can write as $pa-b=0$. Now, we can find $s\not\in M$ such that $sp=q\in L$ and we get an equation, $qa-sb=0$. Since $s\not\in M$, we have proved our claim.