transferring undecidability
A reduction is only useful because something rides along it. What rides along is solvability and its absence. The transfer rule is the engine: a mapping reduction carries decidability forward from the target to the source, and therefore carries undecidability backward from the source to the target. It is like a one-way pipe: pour in 'B is solvable' at one end and 'A is solvable' comes out; equivalently, push 'A is impossible' from the other side and it forces 'B is impossible'.
State it crisply. Suppose A <=m B by a computable function f. Then the two transfer facts are: (the positive direction) if B is decidable, then A is decidable, because to decide A you compute f(w) and run B's decider on it; and (the contrapositive, the one you actually use for hardness) if A is undecidable, then B is undecidable. The second is just the first read by contraposition: if B had a decider, A would too, but A does not, so B cannot. This is why a single seed of undecidability, the halting problem, blossoms into an endless gallery: each new reduction A <=m B with A undecidable instantly stamps B undecidable.
Two honest cautions. First, the transfer only works in the proven direction: A <=m B with A undecidable gives B undecidable; it does NOT tell you B is hard if you reduce the wrong way. Second, transfer carries the property, not a method: knowing B is undecidable because A is undecidable does not hand you any procedure, since neither has one. The reduction is a logical lever, not an algorithm. The very same transfer machinery, with 'decidable' swapped for 'polynomial-time', reappears as the backbone of NP-hardness.
Since A_TM is undecidable and A_TM <=m HALT (map (M, w) to a machine that on input w runs M on w and loops if M rejects), undecidability transfers: HALT is undecidable. The same single reduction, read forward, would say: if HALT were decidable, so would A_TM be.
Decidability flows target-to-source; undecidability flows source-to-target along the same reduction.
Transfer is the contrapositive of the positive rule, so it only licenses the conclusion in one direction. A <=m B with A undecidable proves B undecidable; it never proves A is hard from B's hardness.