First, we present the basic definition, such as that of a ring and a module. We construct a number of theorems needed in proof of the fundamental theorem, including the Chineese remained theorem for rings. Next, we prove two forms of the fundamental theorem of finitely generated modules over PID. Lastly, we employ the module fundamental theorem to prove the fundamental theorem of finitely generated Abelian groups and the existance of the Jordan normal matrix form.
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