I just wondered if someone could verify my calculation that the tensor product of Z + Z/3 and Z/6 is isomorphic to Z/6 + Z/3.
The calculation is as follows:
(Z + Z/6) tensor Z/3 = (Z tensor Z/6) + (Z/3 tensor Z/6) = Z/6 + (Z/3 tensor Z/6)
Now, look at the exact sequence 0 -> Z -> Z -> Z/3 -> 0 with the homomorphisms f: Z -> Z, f(x)=3x and inclusion from Z to Z/3.
Tensor the sequence with Z/6. We get the exact sequence
0 -> Z/6 -> Z/6 -> Z/3 tensor Z/6 -> 0
so Z/3 tensor Z/6 is isomorphic to (Z/6)/f(Z/6) = (Z/6)/(Z/2) = Z/3, hence
Z/6 + (Z/3 tensor Z/6) = Z/6 + Z/3.