## OnLine

### Definitions

abelian
alternating
cyclic
dihedral
finite
general linear
nilpotent
of permutations
of quaternions
order of
projective special linear
simple
solvable
special linear
symmetric
transitive
group algebra

group ring

holomorph (of the integers mod n)

homomorphism, of groups

homomorphism, of modules

homomorphism, of rings

ideal

idempotent element, of a ring

image, of a function

index of a subgroup

injective module

inner automorphism, of a group

integral closure

integral domain

integral extension

integrally closed domain

invariant subfield

inverse function

invertible element, in a ring

irreducible element, in a ring

irreducible polynomial

isomorphism, of groups

isomorphism, of rings

kernel, of a group homomorphism

kernel, of a ring homomorphism

Krull dimension

least common multiple, of integers

left ideal

Legendre symbol

linear action

localization at a prime ideal

maximal ideal

maximal submodule

minimal polynomial

minimal submodule

module

Moebius function

monic polynomial

multiple, of an integer

multiplicity, of a root

nil ideal

nilpotent element, of a ring

nilpotent ideal

Noetherian module

Noetherian ring

normal extension

normal subgroup

normalizer, of a subgroup

one-to-one function

onto function

odd permutation

orbit

order of a group

order of a permutation

p-group

partition of a set

perfect extension

permutation

permutation group

primary ideal

primitive polynomial

principal left ideal

product, of polynomials

projective module

polynomial

prime ideal, of a commutative ring

prime ideal, of a noncommutative ring

prime module

prime number

prime ring

primitive ideal

primitive ring

principal ideal

principal ideal domain

quaternions

regular element

relatively prime integers

right ideal

ring

ring of differential operators

root of a polynomial

root of unity

semidirect product

semiprime ideal

semiprime ring

semiprimitive ring

semisimple Artinian ring

simple extension

semisimple module

separable polynomial

separable extension

simple group

simple ring

simple extension

simple module

skew field

small submodule

socle of a module

split homomorphism

splitting field

stabilizer

subfield

subgroup

subring

Sylow subgroup

symmetric group

tensor product

torsion module

torsionfree module

transcendental element

transposition

unique factorization domain

unit, of a ring

von Neumann regular ring

well-ordering principle

zero divisor

### Theorems

Artin-Wedderburn theorem(11.3.2)

Artin's lemma(8.3.4)

Baer's criterion for injectivity(10.5.9)

Burnside's theorem(7.2.8)

Cauchy's theorem(7.2.10)

Cayley's theorem(3.6.2)

Characterization of free modules(10.2.3)

Characterization of prime ideals(11.1.3)

Characterization of subrings(5.1.3)

Class equation(7.2.6)

Class equation (generalized)(7.3.6)

Cohen's theorem(12.4.1)

Properties of Dedekind domains(12.1.4)

DeMoivre's theorem(A.5.2)

Disjoint cycles commute(2.3.4)

Existence of finite fields(6.5.7)

Existence of maximal submodules(10.1.8)

Existence of quotient fields(5.4.4)

Existence of tensor products(10.6.3)

Euclidean algorithm for polynomials(Example 4.2.3)

Euler's theorem(1.4.11)

Euler's theorem(Example 3.2.12)

Euler's criterion(6.7.2)

Every PID is a UFD(9.1.12)

First isomorphism theorem(7.1.1)

Fitting's lemma for modules(10.4.5)

Frattini's argument(7.8.5)

Fundamental theorem of algebra(8.3.10)

On Galois groups(8.4.3, 8.4.4)

Gauss's lemma(4.3.4)

Hilbert basis theorem(10.3.7)

Hilbert's nullstellensatz(12.4.9)

Hopkin's theorem(11.3.5)

Insolvability of the quintic(8.4.8)

Irreducible ideals are primary(12.3.6)

Jacobson density theorem(11.3.7)

Jordan-Holder theorem for groups(7.6.10)

Kronecker's theorem(4.4.8)

Krull's theorem(12.4.6)

Krull-Schmidt theorem(10.4.9)

Lagrange's theorem(3.2.10)

Maschke's theorem(10.5.8)

Moebius inversion formula(6.6.6)

Nakayama's lemma(11.2.8)

Number of roots of a polynomial(4.1.12)

Order of a permutation(2.3.8)

Partial fractions(Example 4204)

Every p-group is abelian(7.2.9)

Prime and maximal ideals(5.3.9)

Rational roots(4.3.1)

Remainder theorem(4.1.9)

Schur's lemma(10.1.11)

Second isomorphism theorem(7.1.2)

Simplicity of PSL(2,F)(7.7.9)

On solvable groups(7.6.7, 7.6.8)

Splitting fields are unique(6.4.5)

Subgroups of cyclic groups(3.5.1)

Sylow's theorems(7.4.1, 7.4.4)

Wedderburn's theorem(8.5.6)

THE END!