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Answer to Question #16886 in Abstract Algebra for Mohammad
Question #16886
Is Z12 semisimple Z-module?
Expert's answer
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
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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