Commutative ring with a multiiplicative identity, in which every nonzero element is a unit(multiplicative inverse); every field is an integral domain.

*Thm-Finite integral domains are fields

*unity(identity)

Cosets (GIVEN)

*usually not subgroups

Let G be a goup and H a subgroup of G.  Let a exist in G.

aH={ah|hEH}

aH-is Left Coset (a subset of G) of the form

aH={ah|hEH}

Ha-Right Coset

*aH need not be the same as Ha

*Analogously: aHa^-1={aha^-1|hEH}

If H is a subgroup of G, then

|H|\|G|

the order of H divides the order of G

Corr:  Let aEG, then |a|\|G|

Corr:  If |G| is a prime #, then a^|G |=e

Normal Subgroups

If G is a group and if H is a subgroup of G, then

Hi s a normal subgroup of G if for every aEG

aH=Ha

Notation: H triangle G

H triangle G if for every xEG, we have

 xHx^-1 is contained in H

If H is a normal subgroup of G, then the set of all left cosets of H in G form a group where the operation is

(aH)(bH)=(ab)H

written G/H

If Φ^-1-->G′ is a homomorphism and if H′ triangle G′, then

Φ^-1(H) triangle G 

If Φ: G-->G′ is onto, then

G/KerΦ is isomorphic to G′

More Generally,

G\kerΦ is isomorphic to Φ(G)

Given a,b EG,

either

aH=bH

OR

aH union bH = null

Let Z(G) denote the center of G

i)  Z(G) triangle G

ii)  G/Z(G) is isomorphic to Inn(G)

iii)  If G/Z(G) is cyclic, then G is abelian

If G is a group and if H,K are normal subgroups of G with HK=G and H union K = {e},

then G is said to be an internal direct product of H and K.

If G is an internal direct product of H,K, then

G is isomorphic to H + K or H x K

If H,K are groups, then

H+K={(h,k)|hEH,kEK}

with (h1,k1)(h2,k2)=(h1k1,h2k2)

A ring is a set R with 2 binary operations, typically called addition and multiplication satisfying:

1) a+b=b+a (associative)

2) (a+b)+c=a+(b+c)

3) There exists an element 0, such that every element aER a+0=a (identity for addition)

4) for each aER, there exists -aER such that a+-a=0 (inverse)

5) (ab)c=a(bc)

6) a(b+c)=ab+ac or (b+c)a=ba+ca

(distributive)

*We don't assume R has a multiplicative identity

*R might not have a multiplicative inverse

*don't assume ab=ba

If ab=ba, then R is said to be a commutative ring

or if there is a multiplicative identity

If R is a ring with a multiplicative identity, then

an element aER is called a unit if it has a multiplicative inverse. 

Note:  If ab=ac, we can't just conclude that b=c.

If R is a ring and if S is contained in R, then

S is said to be a subring of R if S has all properties of a ring using the same operations as in R.

Integral Domain (GIVEN)

and

Zero Divisor

Zero Divisor:  Let R be a commutative ring.  Let aER with b not equal to 0 and with ab=c, then a is said to be a zero divisor. (2 nonzero things multiplied make zero)

Integral Domain:  A commutative ring which has a multiplicative identity and has NO zero divisors. (well-behaved rings)

Integral Domain Theorem

Cancellation

Let R be an integral domain,

Let a,c,b ER,

with a not equal to 0.

If ab=ac, then b=c.

Characteristic of R

Char R

and

2 Thms

Let R be a ring, the characteristic of R is defined to be the smallest positive integer n, such that na=0 for every aER.  If there is no such integer, the the characteristic is 0.

2 Thms:

1)  If R is a ring with a multiplicative identity, then Char R= order of 1.  If the order is infinite, then the Char of R =0 (order means order under addition).

2)  If R is an integral domain, then Char R=0 or Char R = p where p is a prime.

R/I is an integral domain if and only if I is prime.

R/I is a field if and only if I is a maximal ideal.

An ideal, I is a prime if a,bEI therefore,

aEI or bEI

An ideal I contained in R, is a subring such that if aER, fEI, then

af is contained in I

and

fa is contained in I

**I is an ideal if IR is contained in I

and

RI is contained in I

I is an ideal if:

1)a,bEI therefore a-b EI

2) aEI and rER therefore ar EI and raEI.

Factor Ring

*factor group is a group and factor ring is a ring

If R is a ring and I is an ideal, then

R/I={a+I|aER} forms a ring where

(a+I)+(b+I)=(a+b)+I

(a+I)(b+I)=(ab)+I

A nonempty subset S of a ring R is a subring if S is closed under subtraction and multiplication--that is, if

a-b

and

ab are in S whenever a and b are in s.

Let G be a nonempty set with a binary operation (usually multiplication) that assigns to each ordered pair (a,b) of elements of G an element G denoted by ab. 

G is a group if:

1) (ab)c=a(bc) associativity

2) a(e)=(e)a=a identity

3) ab=ba=e inverse

*if ab=ba, then it is abelian.

Order of a Group

The number of elements of a group (finite or infinite)

denoted as |G|

Let G be a group and let H be a subset of G if

1) a,b EH therefore ab EH

2) aEH therefore a^-1EH

If G is a group, then the center of G is

Z(G)={aEG|ab=ba for all bEG}

*identity

*is a subgroup of G

Let G be a group, let aEG.  The centralizer of a is

C(a)={bEG|ab=ba}

*is a subgroup of G

Let G be a group.  G is said to be cyclic if there exists an element aEG such that <a>=G.

Then a is said to generate G.

If G is a group and if H<_ a, then H is a subgroup of a iff:

1) a,b EH therefore abEH

2) aEH therefore a^-1EH

Four steps to prove G is isomorphic to group G′:

1"Mapping" Define a candidate for the isomorphism that is define a function Φ from G to G′

2. "1-1" Prove that Φ is one to one; that is assume that Φ(a)=Φ(b) and prove a=b.

3.  "onto" Prove that Φ is onto; that is, for any element g in G such that  Φ(g)=g′

4.  "operation perservation" Prove that Φ is operation perserving, that is Φ(ab)=Φ(a)Φ(b)

Kernel of a Homomorphism

The kernel of a homomorphism Φ from a group G to a group with identity e is the set

{xEG|Φ(x)=e}

Let H be a subgroup of G, and let a and b belong to G.  Then,

1. aEaH

2. aH=H iff aEH

3.  aH=bH for Ah union bH = null

4.  aH=bH iff a^-1bEH

5.  |aH|=|bH|

6.  aH=Ha iff H=aHa^-1

7.  aH is a subgroup of G iff aEH.

For every integer a and every prime p,

a^pmodp = amodp

Theorem Classification of Groups of Order 2p