Ongoing deck, Billy Cole

Question 0

what is the ring Z_n

Answer 0

the ring of integrers modulo , with addition and multiplication modulo n.

Question 1

define a Ring. and its 4 conditions.

Answer 1

a ring is given by the following: a set R and two binary operations, +, x, on R. Conditions are;
R1. (R,+) is an abelian group with e=0
R2. x is associative
R3. x is distributive over +
R4 there exists an identity for x, that is not 0

Question 2

lemma 1.3. state the Ring rules.

can you prove each one?

Answer 2

i, 0a=a0=0
ii, a(-b) = (-a)b = -(ab)
iii, (-a)(-b) = ab

Question 3

Lemma 1.5. If S⊆R then S is a subring of R iff ?

Answer 3

1∈ S
r,s ∈ S ⇒ r+s, rxs ∈ S
r ∈ S ⇒ -r ∈ S

Question 4

define a subring.

Answer 4

R = a ring and S⊆R. 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.

Question 5

definition 1.8. define Polynomial Rings.

Answer 5

let R be a ring. the ring of polynomials R[X] in the indeterminate X is defined as follows: 
Elements. ∑ ( a_iX^i )
Equality
Addition
Multiplication

Question 6

what is the degree of a polynomial?

Answer 6

for ∑ ( a_iX^i ), we define,

deg(f) = the largest i such that A_i ≠0

Question 7

what is the deg(f=0) ?

Answer 7

minus infinity

Question 8

formula for deg(fg) ?

Answer 8

deg(fg) ≤ deg(f) + deg(g)

Question 9

formula for deg(f+g) ?

Answer 9

deg(f+g) ≤ max(deg(f),deg(g))

Question 10

def 2.1. define the characteristic, char(R)

Answer 10

the characteristic of a ring R is the least positive integer n such that 1+…+1 =0. that is such that n∙ 1=0.

Question 11

lemma 2.2. suppose that char(R) = n>0 then ?

can you prove the two?

Answer 11

i) n∙ r=0 for every r∈R

ii) if m is a positive integer then m∙ 1=0. iff n divides m.

Question 12

definition2.3. define the domain.

Answer 12

the ring R is a domain if, for all r,s ∈ R

rs=0 ⇒ r=0 or s=0.

Question 13

define a zero divisor

Answer 13

a non-zero element r∈R is a zero divisor if there is a non-zero element s∈R with rs=0 or sr=0. so a domain is a ring without any zero divisors.

Question 14

what is a integral domain?

Answer 14

an integral domain is a commutative domain.

Question 15

lemma 2.5. S⊆R and R is a domain. then ?

Answer 15

S is also a domain, can you prove this? easy

Question 16

♕

Answer 16

♛

Question 17

propostion 2.7.

if R is a domain, then what is R[X]?

Answer 17

also a domain. can you prove this?

Question 18

corollary 2.8.

R is a domain. then the ring R[x_1,…,X_n] is what?

Answer 18

also a domain, as long as R[x_1,…,X_n] is a ring of polynomials in n in-determinates and with coefficients in R.

Question 19

definition 2.9.

a division ring ?

Answer 19

a division ring is a ring in which every non zero element has a right inverse and a left inverse: for every r∈R there is s∈R s.t. rs=1. and there is t∈R s.t. tr=1.

Question 20

define a field

Answer 20

a field is a commutative division ring

Question 21

Lemma 2.11.

R=ring and r∈R has both right and left inverse then ??

Answer 21

these are equal

can you prove ?

Question 22

lemma 2.12.

what are the following conditions on the interger n≥2 that are equivalent.

Answer 22

1. ℤ_n is an integral domain
2. ℤ_n is a field
3. n is a prime

Question 23

Proposition 2.14.

every ?? is a ??. and every ?? is an ??

Answer 23

every division ring is a domain.

and every field is an integral domain

Question 24

lemma 2.16 | in any ring R. the set of units (inverses) R* forms a ???

Answer 24

in any ring R. the set of units (inverses) R* forms a group under multiplication

Question 25

definition 2.17 | define nilpotent

Answer 25

an element r∈R is nilpotent if there is some integer n≥1 with r^n =0.

Question 26

define the index of nilpotence

Answer 26

the least such n that r^n =0.

Question 27

define idempotent

Answer 27

an element r∈R is idempotent if r²=r

Question 28

✎✏✐✏✎✏✐✏✎✏✐✎✏✐✏✎✏✐✏✎✏✐

Answer 28

✎✏✐✏✎✏✐✏✎✏✐✎✏✐✏✎✏✐✏✎✏✐

Question 29

definition 3.1 | define isomorphism

Answer 29

if R&S are rings then an isomorphism from R to S is a bijection θ:R↦S such that, for all r,r'∈R we have θ(r + r') = θ(r) + θ(r') θ(r x r') = θ(r) x θ(r') the left operations are in R and the right operations are in S

Question 30

define R and S are isomorphic

Answer 30

R and S are isomorphic, is there is an isomorphism from R to S

Question 31

lemma 3.3 | suppose that θ:R↦S is an isomorphism. then ....?

Answer 31

1. θ(1)=1 2. θ(0)=0 3. θ(-r) = -θ(r) for every r in R 4. r∈R is invertible iff θ(r)∈S is invertible and, in that case, (θ(r))^-1 = θ(r^-1) 5. r∈R is nilpotent iff θ(r) is nilpotent. (they then have the same index of nilpotence )

Question 32

definition 3.6 | define homomorphism

Answer 32

If R and S are rings then a homomorphism from R to S is a map θ:R↦S such that, for all r,r'∈R we have, θ( r+r' ) = θ(r) + θ(r') θ( rxr' ) = θ(r) x θ(r') we also require that θ(1_R)=1_S

Question 33

lemma 3.7 | suppose that θ:R↦S is an homomorphism. then ....?

Answer 33

1. θ(0)=0 2. θ(-r) = -θ(r) for every r in R 3. if r∈R is invertible then θ(r)∈S is invertible and, in that case, (θ(r))^-1 = θ(r^-1) 4. if r∈R is nilpotent then θ(r) is nilpotent. 5. the image of θ is a subring of S.

Question 34

what is an monomorphism (or an embedding)

Answer 34

it ius an injective (one to one) homomorphism

Question 35

lemma 3.10a | if θ:R↦S and β:S↦T are homomorphisms then ???

Answer 35

so is the composition βθ:R↦T.

Question 36

lemma 3.10b | if θ:R↦S and β:S↦T are embeddings then ??

Answer 36

so is the composition βθ:R↦T.

Question 37

lemma 3.10c | if θ:R↦S and β:S↦T are homomorphisms and if βθ:R↦T is an embedding then ???

Answer 37

θ is an embedding

Question 38

lemma 3.12 | if θ:R↦S is a homomorphism then θ is ???

Answer 38

injective iff ker(θ) = {0}

Question 39

definition 3.11 | define the kernel

Answer 39

if θ:R↦S is a homomorphism of rings then the kernel of θ, ker(θ), is the set {r∈R | θ(r)=0}, of elements which θ sends to 0_S.

Question 40

define an automorphism.

Answer 40

an automorphism of a ring is an isomorphism from the ring to itself. ie the identity map is always an automorphism.

Question 41

lemma 3.17a | suppose that θ:R↦S is a homomorphism. then the ker(θ) is ???

Answer 41

a subgroup of (R,+)

Question 42

lemma 3.17b | let r,r'∈R. then ??? iff ??? iff ????

Answer 42

let r,r'∈R. then θ(r)=θ(r') iff r-r' ∈ker(θ) iff r and r' belong to the same coset of ker(θ) in R

Question 43

definition 3.18 | define an ideal

Answer 43

an ideal of a ring R is subset I⊆R such that: 0∈I a,b ∈I ⇒ a+b∈I a∈I and r∈R ⇒ ar∈I and ra∈I

Question 44

definition 3.19 | define the principle ideal generated by

Answer 44

cant be bothered to write this out

Question 45

Define a principle ideal

Answer 45

An ideal that can be generated by a single element

Question 46

define the trivial ideal

Answer 46

in every ring = {0} is the smallest ideal.

Question 47

define the proper ideal.

Answer 47

in every ring =R is the largest ideal and every other ideal is referred to as a proper ideal

Question 48

define a right ideal

Answer 48

same definition as an ideal but with a weaker third condition; a∈I and r∈R ⇒ ar∈I. left ideals are defined similarly

Question 49

define the principle right ideal generated by

Answer 49

a∈R is defined to be the set {ar | r∈R} and is denoted aR

Question 50

proposition 3.22 | a commutative ring R is a field iff ?

Answer 50

the only ideals of R are {0} and R

Question 51

proposition 3.24 | if θ:R↦S is a homomorphism of rings then ker(θ) ???

Answer 51

is an ideal of R

Question 52

corollary 3.25 | if θ:R↦S is a homomorphism of rings and R is a field then ???

Answer 52

then θ is a monomorphism

Question 54

proposition 3.26 | suppose that I and J are ideals of the ring R, then ??

Answer 54

1. I + J ={a+b | a∈I, b∈J} is an ideal 2. I⋂J is an ideal 3. if {I_λ}_λ is any collection of ideals of R then their intersection ∩_λI_λ is an ideal

Question 55

Factor Rings

Answer 55

U WOT MATE !?!

Question 56

Define the set of cosets of I in the additive group . | Is this definition well defined?

Answer 56

``` R/I = {r + I | r∈R} under operations; ( r + I ) + ( s + I) = ( r + s ) + I ( r + I ) x ( s + I) = ( r x s ) + I Lemma 4.2- the operations are well defined, can you prove it ? ```

Question 57

Lemma 4.3. Let R be a ring and let I be a proper ideal. the set R/I is a ring. what is the the zero element? what is the identity element? what is the inverse ? (provided r has an inverse)

Answer 57

The zero element is the coset 0+I (=I) The identity element is 1+I - ( r + I ) = ( -r ) + I and if r has an inverse in R then ( r + I ) is invertible in R/I with inverse r^-1 + I.

Question 58

Define the Factor Ring of R by I.

Answer 58

``` The set of cosets R/I = {r + I | r∈R}. under operations; ( r + I ) + ( s + I) = ( r + s ) + I ( r + I ) x ( s + I) = ( r x s ) + I with The zero element is the coset 0+I (=I) The identity element is 1+I - ( r + I ) = ( -r ) + I and if r has an inverse in R then ( r + I ) is invertible in R/I with inverse r^-1 + I. ```

Question 59

State the Fundamental Isomorphism Theorem

Answer 59

I = proper ideal of ring R (i) the map π: R↦R/I, defined by π(r) = r + I, is a surjective ring homomorphism with kernal I. (ii) if θ:R ↦ S is a homomorphism and I⊆ker(θ) then there is a unique map θ' : R/I ↦ S with θ'◦π = θ. θ' is a homom (iii) the map θ' is injective iff ker(θ) = I. If θ is surjective and ker(θ) = I then θ' is an isomorphism

Question 60

thm 4.12. if I≤J are ideals of R, so J/I is an ideal of R/I, then??

Answer 60

(R/I)/(J/I) ≃ R/J

Question 61

thm 4.8. let I be an ideal of the ring R. Then there is a natural, inclusion-preserving bijection between the set of ????

Answer 61

of ideals of R which contain I and the set of ideals of the factor ring R/I.

Question 62

Define Maximal

Answer 62

an ideal I of a ring is maximal if it is proper and of there is no ideal between it and R. if any ideal J with I≤J≤R, then J=I or J=R.

Question 62

corollary 4.11. if R is a commutative ring then an ideal I◅R is maximal iff ??

Answer 62

the factor ring R/I is a field.

Question 64

Define a prime proper ideal

Answer 64

a proper ideal I of a commutative ring R is prime whenever r,s ∈R and rs ∈I then either r ∈I or s ∈I

Question 65

POLYNOMIAL RINGS AND FACTORISATION ⚢ + ♂ = ?

Answer 65

GREAT FUN

Question 66

thm 5.1. State the division theorem for polynomials. | what are the quotient and remainder

Answer 66

let K be a field and take f,g ∈K[X] with g≠0. then there are unique q,r ∈K[X] with f=qg + r and deg(r) < deg(g) or r = 0. q is the quotient and r is the remainder when f is divided by g.

Question 67

Define a Root (of f)

Answer 67

An element a ∈ K is a root ( or a zero) of f ∈K[X] if f(a) = 0

Question 68

Corollary 5.4. ?? roots

Answer 68

K is a field. f ∈K[X] and a ∈K. then a is a root of f iff X - a is a factor of f.

Question 69

Define the Greatest common divisor (or HCF) of polynomials.

Answer 69

is a polynomial d such that d divides f and g and if h is any polynomials dividing both f and g then h divides d. write d=gcd(f,g). this polynomial is defined only up to a non-zero scalar multiple so, if we want a unique gcd then we can insist that d be monic.

Question 70

Describe the Division algorithm

Answer 70

-

Question 71

Corollary 5.6. gcd

Answer 71

Let K be a field and take f,g ∈ K[X]. Then the ideal | generated by f and g equals the ideal generated by their greatest common divisor:

Question 72

Corollary 5.8.

Answer 72

Let K be a field. Then every ideal of K[X] is principal (i.e. can be generated by a single polynomial).

Question 73

Define Irreducible

Answer 73

An element r ∈ R is irreducible if r is not invertible and | if, whenever r = st either s or t is invertible.

Question 74

Define Associated

Answer 74

Elements r; s ∈ R are associated if s = ur for some invertible element u ∈ R: For instance two integers r; s are associated iff r = ±s.

Question 75

Define a Unique Factorisation Domain

Answer 75

A commutative domain R is said to be a unique factorisation domain if every non-zero, non-invertible element of R has an essentially unique factorisation as a product of irreducible elements.

Question 76

Define a Principal Ideal Domain

Answer 76

A principal ideal domain, is a commutative | domain in which every ideal is principal (that is, is generated by some single element).

Question 77

Lemma 6.2. | Let K be a field and let f ∈K[X] be irreducible. Then..

Answer 77

is a maximal ideal of K[X].

Question 78

Thm 6.3. State Kronecker's Theorem

Answer 78

Let K be a field and let f ∈K[X] be irreducible of degree n. Define L = K[X]/ then; (i) L is a field and homomorphism π: K[X] ↦ K[X]/ induces an embedding θ: K↦L. (ii) α= π(X)∈L is a root of f (iii) The dimension of L as a vector space over K is n, with {1,α,α^2,..., α^n-1 } being a basis of L over K, so every element of L has a unique representation of the form a_n-1α^n-1 + ... + a_1α + a_0 with a_n-1,...a_0 ∈K

Question 79

Define Algebraically Closed

Answer 79

A field K is said to be algebraically closed if any polynomial f ∈K[X] has all its root in K.