Exercises in Classical Ring Theory

acts algebra artinian assume central character claim clearly commutative conclusion conjugate consider constructed contains contradiction Conversely course defined desired direct division ring domain easy element equation equivalent example Exercise exists extension fact field finite follows formally given gives group G hence holds homomorphism idempotent identity implies infinite instance integer inverse irreducible isomorphism left primitive left R-module linear matrix maximal left ideal means minimal module multiplication Neumann regular nilpotent noetherian nonzero Note particular polynomial prime ideal problem projective proof properties prove R-module radical rank reduced relations representation resp result right ideal root satisfies semilocal semiprime semisimple semisimple ring Show simple Solution strongly subgroup submodule subring suffices Suppose Theorem theory unique unit write zero