Let $\mathbb A_K$ be the additive group of adeles associated to a number field $K$.
I suspect that there is no surjective group homomorpphism $\phi:\mathbb A_K\to K$ such that the kernel is of the type $b\mathbb A_K$ for $b\in \mathbb A_K\setminus 0$. Do you have any idea how to prove (or disprove) this? In particular, given an element $(a_\mathfrak p)\in \mathbb A_K$ I don't see any way to produce an element $f\in K$ that respects the sum.