Integral Domains - ksuweb.kennesaw.eduksuweb.kennesaw.edu/~plaval/math4362/slides_intdom.pdf1 An...

Integral Domains

Philippe B. Laval

KSU

Current Semester

Philippe B. Laval (KSU) Integral Domains Current Semester 1 / 14

Introduction

In this chapter, we study properties of integral domains. In the nextchapter, we will focus on the ring of integers.

An integral domain is a good abstraction for the ring of integers.

We will define and study the notion of characteristic of a ring andsee how it applies to integral domains.

The characteristic of a ring is related to the notion of order of anelement in a group.


Introduction

Before proceeding, it will be convenient to introduce some useful notationwhich applies to any ring, not just integral domains. If R is any ring,a ∈ R and n a positive integer. We define:

1 n · a =a+ a+ a+ ...+ a︸︷︷︸

n times2 0 · a = 03 (−n) · a = − (n · a)

With these definitions, we have the following useful algebraic rules. If R isany ring, a ∈ R and m, n positive integers, we have:

1 ma+ na = (m + n) a2 m (na) = (mn) a


Definitions

Recall that an integral domain can be defined two ways. These two waysare equivalent, as we have already proven.

Definition (Integral Domain)Two equivalent definitions of an integral domain:

1 An integral domain is a commutative ring with unity having nodivisors of zero that is ab = 0 =⇒ a = 0 or b = 0.

2 An integral domain is a commutative ring with unity having thecancellation property that is if a 6= 0 and ab = ac then b = c .


Examples of Integral Domains

ExampleThe ring of Gaussian integers, Z [i ] = {a+ bi : a, b ∈ Z}.

ExampleThe ring of polynomials in the variable x with integer coeffi cients, Z [x ].

Example

Z[√2]={a+ b

√2 : a, b ∈ Z

}ExampleZp , the ring of integers modulo p when p is prime.


Examples Which Are Not Integral Domains

ExampleZn, the ring of integers modulo n when n is not prime. It has divisors ofzero.

ExampleThe ring M2 (Z), the set of 2× 2 matrices with integer entries. It is anon-commutative ring with unity.


Important Property

The only thing missing for an integral domain to be a field is the existenceof a multiplicative inverse for each non-zero element. It turns out thatfinite integral domains do have this property and hence are fields.

TheoremA finite integral domain is a field.

Sketch of a proof:

1 Let D be a finite integral domain with n elements .

2 Write D = {0, 1, a1, a2, ..., an−2}. Let ai ∈ D, show a−1i ∈ D.3 Consider the elements ai0, ai1, aia1, aia2, ...How many distinct suchelements are there?

4 Finish the proof.

ExampleZp , the ring of integers modulo p is a field if p is prime.


Important Property



Sketch of a proof:

1 Let D be a finite integral domain with n elements .2 Write D = {0, 1, a1, a2, ..., an−2}. Let ai ∈ D, show a−1i ∈ D.

3 Consider the elements ai0, ai1, aia1, aia2, ...How many distinct suchelements are there?

4 Finish the proof.



Important Property



Sketch of a proof:

1 Let D be a finite integral domain with n elements .2 Write D = {0, 1, a1, a2, ..., an−2}. Let ai ∈ D, show a−1i ∈ D.3 Consider the elements ai0, ai1, aia1, aia2, ...How many distinct suchelements are there?

4 Finish the proof.



Characteristic of a Ring

Recall, in a group, the order of an element a is defined to be thesmallest positive integer n such that na = 0 (additive notation).

To emphasize the fact that we are referring to the order of a in termsof addition, we will call it the additive order. In a ring R, (R,+)forms an Abelian group. Hence, we can talk about the additive orderof an element.

We now define the notion of characteristic of a ring and see how itapplies to integral domains. It will be defined in terms of additiveorder.

Definition (Characteristic of a Ring)The characteristic of a ring R is the least positive integer n such thatnx = 0 for all x in R. If no such integer exists, we say that R hascharacteristic 0. The characteristic of R is often denoted by char (R).


Examples

Examplechar (Z) = char (Q) = char (R) = char (C) = 0.

ExampleLet A = {0}, the trivial ring. Then char (A) = 1.


Characteristic of an Integral Domain

RemarkThe above implies that if D is an integral domain and char (D) = n > 0then n · x = 0 for every x in D.

In the case of an integral domain, the definition is much simpler.

TheoremIf D is an integral domain then all the non-zero elements have the sameadditive order.

Sketch of a proof:

1 Let a ∈ D with a 6= 0

2 Show that na = 0⇐⇒ (n · 1) = 03 Finish the proof






Sketch of a proof:

1 Let a ∈ D with a 6= 02 Show that na = 0⇐⇒ (n · 1) = 0

3 Finish the proof






Sketch of a proof:

1 Let a ∈ D with a 6= 02 Show that na = 0⇐⇒ (n · 1) = 03 Finish the proof



We can now define the characteristic of an integral domain as follows:

DefinitionLet D be an integral domain. If 1 has additive order n, we define thecharacteristic of D to be n. Otherwise, the characteristic of D is 0.

ExampleWhat is the characteristic of Z5?

ExampleShow by example that the theorem does not work for Z6. Why is it?


Properties

In fact, in the case of an integral domain, the possibilities for itscharacteristic are even more limited.

TheoremThe characteristic of an integral domain is 0 or a prime number.

Sketch of a proof:

1 Explain why it is enough to show that if the additive order of 1 isfinite, it must be prime.

2 Suppose that 1 has order n and n = st where 1 ≤ s, t ≤ n. Show thatwe must have either s = n or t = n.

3 Finish the proof.


More Properties

The final result is the wish of every student!

TheoremLet D be an integral domain with char (D) = p > 0, then ∀a, b ∈ D,

(a+ b)p = ap + bp

Sketch of a proof:

1 Recall (a+ b)p = ap +p−1∑k=1

(pk

)ap−kbk + bp . It can be proven that

the binomial formula is correct in every commutative ring.

2 Explain why the binomial coeffi cient(pk

)=p (p − 1) (p − 2) ... (p − k + 1)

k!is a multiple of p when p is

prime.3 Finish the proof.


More Properties



(a+ b)p = ap + bp

Sketch of a proof:


(pk


the binomial formula is correct in every commutative ring.2 Explain why the binomial coeffi cient(

pk

)=p (p − 1) (p − 2) ... (p − k + 1)


prime.

3 Finish the proof.


More Properties



(a+ b)p = ap + bp

Sketch of a proof:


(pk


the binomial formula is correct in every commutative ring.2 Explain why the binomial coeffi cient(

pk

)=p (p − 1) (p − 2) ... (p − k + 1)


prime.3 Finish the proof.


Exercises

Do the following problems at the end of chapter 20 of your book: A1, A2,A3, A6, A7, B1, B2, C2, C3, E1, E4.


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