I think requiring A to be divisible, or even just p^n divisible, for any natural number n, will do the trick: let a (x) b be any basic tensor ==> since b in B and b is a p-group, there exists a natural number n s.t. p^n*b = 0, and also there exists a' in A s.t. a = p^n*a', so:

a (x) b = p^n*a' (x) b = a' (x) p^n*b = a (x) 0 = 0

Tonio