When is the concatenation of two regular languages unambiguous?
Hint: Given DFAs for $A$ and $B$, construct an NFA which accepts words in $AB$ having at least two different decompositions. The NFA keeps track of two copies of the standard NFA for $AB$ (formed by joining DFAs for $A$ and $B$ with $\epsilon$ transitions), ensuring that the switch from $A$ to $B$ happens at two different points.