User:TakuyaMurata/Algebra exercises

An integral domain is a PID if and only if every submodule of a free module over it is free.
Not every module contains a maximal free submodule. (Consider Q as a Z-module?.)
Not every module contains a maximal submodule. Give an example.
Let A be a Noetherian integrally closed domain. Then the dual of a finitely generated module over A is reflexive. (Hint: any module is a quotient of a free module.) Can you weaken the assumption on A?
Discuss: countably generated modules and continuum hypothesis
Discuss: an Artinian ring and the definitions of radicals of a ring.
(a) The group algebra of a reductive algebraic group over a field of characteristic zero is a semisimple algebra. (b) Discuss the positive characteristic case (cf. Maschke's theorem.)

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