Ring Theory

abelian group artinian ring biregular chain condition commutative ring compact congruence relation contradiction decomposition defined Definition denote direct sum direct summand division ring endomorphism epimorphism equivalence essential extension exists factor ring faithful finite number Frobenius algebra Hence Homk(M homomorphism hypothesis idempotent implies indecomposable left ideals injective intersection irreducible R-module isomorphic K-linear lattice left R-module lemma linear mapping matrix maximal ideal maximal left minimal left ideal module monomorphism multiplication nilpotent noetherian nonzero one-one orthogonal partial order polynomial primitive ideals primitive ring Proof prove QD-algebra quotient ring R-ideal R-submodule radical regular element representation right ideal right K-module ring Q ring with identity semigroup semiperfect ring semisimple artinian ring simple ring splitting field subdirect sum submodule subring subset Theorem 12 topological U-irreducible vector space zero