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Transcript of 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.
Philippe B. Laval (KSU) Integral Domains Current Semester 2 / 14
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
Philippe B. Laval (KSU) Integral Domains Current Semester 3 / 14
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 .
Philippe B. Laval (KSU) Integral Domains Current Semester 4 / 14
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.
Philippe B. Laval (KSU) Integral Domains Current Semester 5 / 14
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.
Philippe B. Laval (KSU) Integral Domains Current Semester 6 / 14
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.
Philippe B. Laval (KSU) Integral Domains Current Semester 7 / 14
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.
Philippe B. Laval (KSU) Integral Domains Current Semester 7 / 14
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.
Philippe B. Laval (KSU) Integral Domains Current Semester 7 / 14
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.
Philippe B. Laval (KSU) Integral Domains Current Semester 7 / 14
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).
Philippe B. Laval (KSU) Integral Domains Current Semester 8 / 14
Examples
Examplechar (Z) = char (Q) = char (R) = char (C) = 0.
ExampleLet A = {0}, the trivial ring. Then char (A) = 1.
Philippe B. Laval (KSU) Integral Domains Current Semester 9 / 14
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
Philippe B. Laval (KSU) Integral Domains Current Semester 10 / 14
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= 02 Show that na = 0⇐⇒ (n · 1) = 0
3 Finish the proof
Philippe B. Laval (KSU) Integral Domains Current Semester 10 / 14
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= 02 Show that na = 0⇐⇒ (n · 1) = 03 Finish the proof
Philippe B. Laval (KSU) Integral Domains Current Semester 10 / 14
Characteristic of an Integral Domain
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?
Philippe B. Laval (KSU) Integral Domains Current Semester 11 / 14
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.
Philippe B. Laval (KSU) Integral Domains Current Semester 12 / 14
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.
Philippe B. Laval (KSU) Integral Domains Current Semester 12 / 14
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.
Philippe B. Laval (KSU) Integral Domains Current Semester 12 / 14
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.
Philippe B. Laval (KSU) Integral Domains Current Semester 13 / 14
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.
Philippe B. Laval (KSU) Integral Domains Current Semester 13 / 14
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.
Philippe B. Laval (KSU) Integral Domains Current Semester 13 / 14