1) Introduction to Rings

Question 1

What is the division theorem for ℤ

Answer 1

If you have two integers a and b with b > 0.
Then there exist unique integers q and r such that a = bq + r and 0 ≤ r < b

Question 2

What does it mean for a to divide b

Answer 2

a divides b if and only if there exists an integer k such that b = ka

Question 3

What is the greatest common divisor of two integers

Answer 3

gcd(a, b) = max{g ∈ Z | g divides a and g divides b}

Question 4

When are two integers coprime

Answer 4

If gcd(a, b) = 1

Question 5

What is the Euclidean Algorithm

Answer 5

Use the Division Theorem repeatedly to calculate

Question 6

What does it mean for a to be congruent to b modulo n

Answer 6

a is congruent to b modulo n if and only if n divides
a − b (n ∈ Z with n ≥ 2)

Question 7

What is the congruent class of of a modulo n

Answer 7

Question 8

What is the set of all congruence classes mod n

Answer 8

Question 9

What does the Division Theorem imply about the congruence of every integer modulo n

Answer 9

By the Division Theorem every integer is congruent to a unique element of {0, 1, 2, …, n − 1} modulo n
Therefore Zn = {[0]n, [1]n, …, [n − 1]n}

Question 10

What is the critera for two integers
a and b to be congruent modulo n

Answer 10

b ≡ a mod n if and only if b - a = kn for some k ∈ Z

Question 11

What is another way we can represent [a]n

Answer 11

[a]n = {a + nk | k ∈ Z}
[a]n = [a + kn]n for all k ∈ Z

Question 12

Let a, b, n ∈ Z with n ≥ 2. Prove that a ≡ b mod n if and only if
[a]n = [b]n

Answer 12

Question 13

What are the properties of congruence modulo n concerning addition and multiplication

Answer 13

Let a, b, c, d, n ∈ Z with n ≥ 2 and a ≡ b mod n and c ≡ d mod n,
Addition: a + c ≡ b + d mod n
Multiplication: ac ≡ bd mod n

Question 14

Describe the proof of the properties of congruence modulo n concerning addition and multiplication

Answer 14

• Assume a ≡ b mod n and c ≡ d mod n.
• Then we can write a = b + rn and c = d + sn for some r, s ∈ Z.
• Then a + c = b + d + (r + s)n and so a + c ≡ b + d mod n.
• We have ac = (b + rn)(d + sn) = bd + n(rd + sb + rsn) and so ac ≡ bd mod n

Question 15

What is a composite number

Answer 15

A number that is not prime

Question 16

What is Euclid’s property of primes

Answer 16

Let p be a prime and a, b ∈ Z. If p divides ab, then p divides a or p divides b

Question 17

Describe the proof of Euclid’s property of primes

Answer 17

Let p be a prime. Assume that p | ab
If p | a, then we are done.
Suppose that p ∤ a. Then p and a are coprime because the only positive factors of p are 1 and p
Using the Euclidean algorithm we can write 1 = am + pn for some m, n ∈ Z.
Therefore b = abm + pbn and since p | ab we have that p | (abm + pbn).
Hence p | b

Question 18

What is the Fundamental Theorm of Arithmetic

Answer 18

Every integer n greater than 1 can be written as a product of primes that is unique up to ordering

Question 19

Describe the proof of the uniqueness of prime factorisation

Answer 19

Question 20

Prove that in Zp, [(p − 1)!]p = [−1]p

Answer 20

• Since p is a prime, every non-zero element of Zp has an inverse
• The only elements of Zp that are self-inverse are [1]p and [−1]p
• Therefore the other non-zero elements pair up as inverses mod p in the product (p − 1)!. Hence [(p − 1)!]p = [1]p[2]p…[p − 1]p = [1]p[p − 1]p = [p − 1]p = [−1]p

Question 21

What is a ring

Answer 21

A ring is a set R with two binary operations, + and ×, on R which satisfies the following conditions:
* (R1) (R, +) is an abelian group with identity 0.
* (R2) × is associative
* (R3) × is distributive over +
* (R4) there exists an element 1 ∈ R, different from 0, that is the identity for ×

Question 22

Is + or x commutative in a ring

Answer 22

• (+ is always commutative in a ring but × may not be commutative
• If × is commutative we call R a commutative ring

Question 23

Describe the multiplication properties of rings

Answer 23

Let R be a ring. Then, for all a, b ∈ R
(i) 0a = a0 = 0
(ii) a(−b) = (−a)b = −(ab)
(iii) (−a)(−b) = ab.

Question 24

Describe the proof of the multiplication properties of rings

Answer 24

(i) We have 0a + 0a = (0 + 0)a = 0a
and so be adding −0a to both sides we get 0a = 0. Likewise a0 = 0
(ii) By definition −(ab) is the unique element in R with the property that ab + (−(ab)) = 0. Now
ab + (−a)b = (a + (−a))b = 0b = 0
Hence (−a)b = −(ab) by uniqueness. Similarly a(−b) = −(ab)
(iii) By using (ii) twice and −(−a) = a we get
(−a)(−b) = −(−(ab)) = ab

Question 25

What is a subring

Answer 25

S is a subring of R if it is a ring in its own right with respect to the same addition and multiplication as in R and S contains 1R

Question 26

What is the subring test

Answer 26

S is a subring of R iff all the following conditions hold
(i) 1R ∈ S
(ii) r + s, r × s ∈ S, for all r, s ∈ S
(iii) −r ∈ S for all r ∈ S

Question 27

Describe the proof of the subring test

Answer 27

Suppose S satisfies the three conditions of the subring test
Let r, s ∈ S. Using the subgroup test, (S, +) is a subgroup of (R, +) by (ii) and (iii)
Commutativity of +, associativity of × and distributivity of × over + are inherited from R
Using (i) and (iii) we have −1R ∈ S and then 0R = 1R + (−1R) ∈ S by (ii)
Since 0R, 1R ∈ S and R ≠ 0R in R, we have 1R ≠ 0R in S. Therefore S satisfies (R1)-(R4) and is a ring

Question 28

Explain why ℚ[√2, √3] is the smallest subring of ℝ containing ℚ, √2 and √3

Answer 28

Because any other subring of ℝ satisfying this condition is
closed under addition and multiplication and therefore contains all elements of the form a + b√2 + c√3 + d√6 (= ℚ[√2, √3])

Question 29

Define the polynomial ring R[X]

Answer 29

Question 30

How is the degree of a polynomial defined

Answer 30

For f = ∑i≥0aiXi we define deg(f) to be the largest i such that ai ≠ 0

Question 31

When does a polynomial have
deg(f) =−∞

Answer 31

If f = 0

Question 32

What are the degree properties for the sum and product of two polynomials

Answer 32

deg(f + g) ≤ max(deg(f), deg(g))
deg(fg) ≤ deg(f) + deg(g)

Question 33

What is a Cartesian Product

Answer 33

If R1 and R2 are rings then the Cartesian product R1 × R2 with operations + and × defined by
(r1, r2) + (s1, s2) = (r1 + s1, r2 + s2) and
(r1, r2) × (s1, s2) = (r1 × s1, r2 × s2) is a ring