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
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