Types of Groups and Rings

Group

• Closed under *

• * is associative

• Has an identity

• Has inverses

Abelian Group

A commutative group

Klein 4-Group

A group with only 4 elements in it, and each non-identity element has order 2

Subgroup

• Closed under the same operation as parent group

• Has an identity

• Has inverses

  • \forall a, b \in H, ab \in H

Cyclic Group

A commutative group which can be generated from one element.

Coset

aH = \{ah:h\in H\} where H\leq G, a\in G

Normal Subgroup

Denoted: H\trianglelefteq G

Let H\subseteq G and ghg^{-1}\forall g\in G, h\in H

Simple Group

A group with no normal subgroups

Quotient/Factor Group

Denoted: G/N

Where N\trianglelefteq G, G/N = \{aN:a\in G\}

(Note: (aN)(bN) = abN, I_{G/N}=N and (aN)^{-1}= a^{-1}N

Internal Direct Product

Let a group G=HK where H\trianglelefteq G, K\triangelefteq G and H\cap K=\{e\}.

(Note: g=hk is unique, hk=kh, G\cong H\times K and G/H=K.

Ring

• (R,+) is an abelian group

• Closed under + and \times

• Associative

• Distributive

• (Has an identity if 1\in R s.t. 1\neq 0 and a\times 1 = a

Commutative Ring

A ring where ab = ba

Subring

S\subseteq R if

• 0\in S

• \forall a, b \in S, -a, a+b and ab \in S

• Closed under the same + and \times as R

Field

A commutative ring with an identity where U(F) = F\backslash\{0\}

Integral Domain

A commutative ring with an identity where ZD(R)=\{0\}

Ideal

I\subseteq R if

• 0\in I

• a\in I \Rightarrow a^{-1}\in I

• a, b \in I \Rightarrow a+b \in I

• \forall a \in I, r\in R \Rightarrow ra, ar \in I

Quotient/Factor Ring

The collection of all distinct cosets of R, R/I where I is an ideal of R.

(Note: + is defined as: (a+I)+(b+I)=(a+b)+I and \times is defined as: (a+I)\times(b+I)=(ab)+I

Direct Product Ring

The cartesian product, R\times S where + is defined by: (r,s)+(r’,s’)=(r+r’, s+s’) and \times is defined by: (r,s)\times(r’,s’)=(rr’,ss’)

Principal Ideal Domain

An integral domain where all the ideals are principal (i.e. generated by a single element)

Unique Factorisation Domain

An integral domain where each element a (\neq 0) can be decomposed into a unique multiplication of prime numbers.

Proper Ideal

An ideal I where I\neq R

Prime Ideal

A proper ideal where if ab\in P then a\in P or b\in P

Maximal Ideal

A proper ideal M where no other I exists such that M\subset I\subset R except when I=M or I=R.