Question

Is Z12 semisimple Z-module?

Answer 1

Z_12=Z_3 +Z_4
Z_3 is simple Z-module but Z_4 is not semisimple. because it
cant be split to the direct sum of simple Z-modules. Why? because all simple
Z-modules looks like Z_p where p is prime. so for Z_4 we have only 1 possibility
Z_4=Z_2 + Z_2. but its not true
so Z_4 is not semisimple