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In each of the following statement, identify if they are true, false or partially true. In case it is true, give proof. In case it is false give counter example with solution. In case it is partially true then give one example each of the case in which it is true and in which it is not true. 1. If R is Artinian, then R[x] is Artinian. 2. A finitely generated module over a PID is Noetherian. 3. Minimal modules exist in Noetherian modules. 4. Maximal Submodule need not exist in an Artinian module. 5. A Noetherian module need not be Artinian. 6. A maximal submodule is always simple.
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