Modern Algebra, Volume 1

a₁ abelian group additive group algebraic structure automorphism b₁ belongs bijection c₁ commutative ring composition Consequently contains COROLLARY coset defined DEFINITION denoted direct sum division ring E₁ endomorphism epimorphism equivalence relation Example Exercise exists greatest common divisor group G hence homomorphism idempotent identity element induction integral domain invertible elements irreducible elements isomorphism K-algebra K-vector K[X₁ kernel left ideal Let ƒ Let G linear operator linear transformation linearly independent m₁ module monomorphism n-dimensional natural numbers neutral element nonempty nonzero elements normal subgroup ordered basis permutation polynomial ƒ principal ideal Proof prove quotient real numbers right ideal ring with identity satisfying scalar multiplication semigroup sk≤n splitting field subalgebra subfield subgroup of G submodule subring subspace surjective Theorem total ordering unitary K-module vector space whence zero zero-divisor