Exercises in Classical Ring Theory

a₁ abelian archimedean artinian ring assume automorphism b₁ central idempotents commutative ring conjugate constructed contradiction cyclic defined division algebra division ring domain element endomorphism equation Exercise exists fact field finite finite-dimensional follows group G hence homomorphism idempotent implies indecomposable integer inverse irreducible isomorphism J-semisimple Jacobson k-algebra kG-module L-rad left ideal left primitive ring Lemma local ring matrix maximal ideal maximal left ideal maximal subfield maximal submodule module multiplication Neumann regular ring nil ideal nilpotent ideal noetherian ring noncommutative nonzero polynomial preordering prime ideal primitive idempotents primitive rings proof prove R-module R/rad rad kG radical resp right ideal right perfect right R-module semilocal ring Semiperfect Rings semiprimary ring semiprime semisimple ring simple ring Solution stable range strongly regular subdirect product subgroup submodule subring suffices to show T-nilpotent Theorem unit-regular zero