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.)