Quotient Structures

 

  Sets Groups Rings Fields
Subsetting term Subset A subgroup H of a group G is called a normal subgroup

if a-1ha is in H for all a in G and h in H.

A nonempty subset I of ring R is called an ideal if

x,y are in I and a is in R and

x-y is in I

x*a and a*x is in I.

 A function is  irreducible if it can't be factored in R.
Equalence class definition Defined by a relation on S.

equivalence class

[a]={x in S | xRa}. The partition S into disjoin (sets that do not have common elements) sets.

The equivalence classes are the orbits of x.

 The sets defined by the relation of the normal subgroup H on group G are called cosets. (R is a relation.)

Ha ={hRa | h in H}and

aH ={aRh |h in H}

 a in G; a is a representitive of coset HRa.

 If I is an ideal in ring R, then the cosets forms a ring {R/I, +,*) definded by

(I+a1) +(I+a2)=I +(a1+a2)

(I+a1)(I+a2)=I+(a1a2)

These are also called the residue class ring with respect to ideal I.

  
Designation S/R={[a] | in S}

The set of all equivelence classes is called a quotient set. R stands for any relation.

(G/H, R)

The set of all cosets is called a quotient group.The R stands for any relation. H is a normal subgroup.

{R/I, +,*)

The set of all cosets is called a quotient ring. The + and * stand for any relation. I is a ideal.

  F/(p(x)

If F is a field and p(x) is irreducible then the set of F/p(x) is a quotient field.
Notes     The zero ideal and the unit ideal are trivial ideas. All others are proper ideals.

The set {ar |r in R} of all multiples of a is an ideal called the principal ideal generated by a and is known by (a). (a)=aZ consists of all integer multiples of a in Z and is the principal ideal generaled by a in Z. If every ideal is principal, the the ring is called a principal ideal ring.

If aZ={ax |x in Z} then the ideal is called the prime ideal. For each a and b in R, if ab is in P then a is in P or b is in P where P is a prime ideal.

An ideal M in ring R is a maximal ideal if and only if M<>R and I is an ideal such that M<I<R, then I=M or I=R. There are no proper ideals.

  
Examples  The set of integers modulo n. The modular arithmethic of (Zn, +). The modular arithmethic of (Zn, +, *).  Zp=Z/(p), prime modular systems.

F[x]/(p(x), polynomial with P(x) as a irreducible factor.

A Galois Field GF(pm)=Zp[x]/(p(x)

R[x]/(x2+1) = {a+bx | a,b in R} This generates the complex numbers.

 

Example of a Quotient Group

Let H={0,4,8} H is a normal subgroup of the group G,   (Z12 ,+)  the integers modulo 12. The cosets (G/H, +) are

H={0,4,8}=4+H=8+H            note: 4+H=H, so there are only 4 subgroups of Z12.
1+H={1,5,9}=5+H=9+H       0,1,2 and 3 are representives of their cosets.
2+H={2,6,10}=6+H=10+H
3+H={3,7,11}=3+H=11+H

 

+ H 1+H 2+H 3+H
H H 1+H 2+H 3+H
1+H 1+H 2+H 3+H H
2+H 2+H 3+H H 1+H
3+H 3+H H 1+H 2+H

 

Example of a Quotient Ring

If {0,2,4} is the ideal generated by 2 in Z6, there is a quotient ring Z6/I.

There are two cosets: I={0,2,4} and I+1={1,3,5}

Hence the quotient ring Z6 is given by Z6/I={I, I+1} = Z6/{0,2,4} = Z2

+ I I+1   * I I+1
I I I+1   I I I+1
I+1 I+1 I   I+1 I I+1

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