I'm currently reading up a bit on Turing decidability reductions as I'll take a more in-depth course about the topic next semester. While reading up on some older course material I asked myself the following question which leaves me a bit puzzled:
Let $A, B$ be languages, and $\le_m$ be the mapping reduction for languages.
For mapping reductions a central statement is
If $A \le _m B$, then $B$ decidable $\implies A$ decidable
I'm now wondering about is whether the implication also works the other way around:
If $A \le _m B$, then $A$ decidable $\implies B$ decidable
I strongly suspect that this is not the case as $A \le _m B$ does not even imply the existence of a mapping function from $B$ to $A$, however I'm struggling to find a good counterexample for this.