A ring is noetherian iff the category of finitely generated -modules is a full abelian subcategory of the category of all -modules.
Most of representation theory results work out for algebraically closed fields. For instance, we can classify irreducible representations for the dihedral group with elements over . Classify all irreducible representations for over .
Let be the cyclic group of order and consider the group algebra. Prove that the number of solutions to the equation for is for .
Let be an algebra over and let be an irreducible representation of with at most countable basis. Then any homomorphism of representations is a scalar operator.
Any field that is finitely generated as a ring is a finite field.
Exhibit a group , a set of generators for and a subgroup such that for all and yet is not normal in .
Is there a ring such that there are two different ring injections ?