Please give your comments on the correctness of the following. Thanks. My question is : Give an example of a Z-module of finite length not being semisimple. I want to show that Z_4 where Z = integers is not simple. 2Z_4 is a subgroup of order 2 which is normal subgroup, hence Z_4 is not simple. But how can I argument that this module is not semisimple? Z_4 can not be written as a direct sum of simple subgroups because the only subgroups of Z_4 are (0), 2Z_4 and Z_4. So we can not write Z_4 as a direct sum of proper subgroups. Is it correct. And this module has finite length since 0/2Z_4 =0 and 2Z_4/Z_4 =0.