Brainscape's Knowledge GenomeTM

Browse over 1 million classes created by top students, professors, publishers, and experts.


See full index

Algebraic Structures 2 Billy Cole > Ongoing deck, Billy Cole > Flashcards

Ongoing deck, Billy Cole Flashcards

0
Q

what is the ring Z_n

A

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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
1
Q

define a Ring. and its 4 conditions.

A

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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
2
Q

lemma 1.3. state the Ring rules.

can you prove each one?

A

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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
3
Q

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

A

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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
4
Q

define a subring.

A

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.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
5
Q

definition 1.8. define Polynomial Rings.

A 
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
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
6
Q

what is the degree of a polynomial?

A

for ∑ ( a_iX^i ), we define,

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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
7
Q

what is the deg(f=0) ?

A

minus infinity

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
8
Q

formula for deg(fg) ?

A

deg(fg) ≤ deg(f) + deg(g)

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
9
Q

formula for deg(f+g) ?

A

deg(f+g) ≤ max(deg(f),deg(g))

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
10
Q

def 2.1. define the characteristic, char(R)

A

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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
11
Q

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

can you prove the two?

A

i) n∙ r=0 for every r∈R

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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
12
Q

definition2.3. define the domain.

A

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

rs=0 ⇒ r=0 or s=0.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
13
Q

define a zero divisor

A

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.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
14
Q

what is a integral domain?

A

an integral domain is a commutative domain.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
15
Q

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

A

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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
16
Q

♕

A

♛

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
17
Q

propostion 2.7.

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

A

also a domain. can you prove this?

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
18
Q

corollary 2.8.

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

A

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.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
19
Q

definition 2.9.

a division ring ?

A

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.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
20
Q

define a field

A

a field is a commutative division ring

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
21
Q

Lemma 2.11.

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

A

these are equal

can you prove ?

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
22
Q

lemma 2.12.

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

A
  1. ℤ_n is an integral domain
  2. ℤ_n is a field
  3. n is a prime
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
23
Q

Proposition 2.14.

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

A

every division ring is a domain.

and every field is an integral domain

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
24
Q

lemma 2.16

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

A

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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
25
Q

definition 2.17

define nilpotent

A

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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
26
Q

define the index of nilpotence

A

the least such n that r^n =0.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
27
Q

define idempotent

A

an element r∈R is idempotent if r²=r

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
28
Q

✎✏✐✏✎✏✐✏✎✏✐✎✏✐✏✎✏✐✏✎✏✐

A

✎✏✐✏✎✏✐✏✎✏✐✎✏✐✏✎✏✐✏✎✏✐

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
29
Q

definition 3.1

define isomorphism

A

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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
30
Q

define R and S are isomorphic

A

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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
31
Q

lemma 3.3

suppose that θ:R↦S is an isomorphism. then ….?

A
  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 )
32
Q

definition 3.6

define homomorphism

A

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

33
Q

lemma 3.7

suppose that θ:R↦S is an homomorphism. then ….?

A
  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.
34
Q

what is an monomorphism (or an embedding)

A

it ius an injective (one to one) homomorphism

35
Q

lemma 3.10a

if θ:R↦S and β:S↦T are homomorphisms then ???

A

so is the composition βθ:R↦T.

36
Q

lemma 3.10b

if θ:R↦S and β:S↦T are embeddings then ??

A

so is the composition βθ:R↦T.

37
Q

lemma 3.10c

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

A

θ is an embedding

38
Q

lemma 3.12

if θ:R↦S is a homomorphism then θ is ???

A

injective iff ker(θ) = {0}

39
Q

definition 3.11

define the kernel

A

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.

40
Q

define an automorphism.

A

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

41
Q

lemma 3.17a

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

A

a subgroup of (R,+)

42
Q

lemma 3.17b

let r,r’∈R. then ??? iff ??? iff ????

A

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

43
Q

definition 3.18

define an ideal

A

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

44
Q

definition 3.19

define the principle ideal generated by

A

cant be bothered to write this out

45
Q

Define a principle ideal

A

An ideal that can be generated by a single element

46
Q

define the trivial ideal

A

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

47
Q

define the proper ideal.

A

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

48
Q

define a right ideal

A

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

49
Q

define the principle right ideal generated by

A

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

50
Q

proposition 3.22

a commutative ring R is a field iff ?

A

the only ideals of R are {0} and R

51
Q

proposition 3.24

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

A

is an ideal of R

52
Q

corollary 3.25

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

A

then θ is a monomorphism

54
Q

proposition 3.26

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

A
  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
55
Q

Factor Rings

A

U WOT MATE !?!

56
Q

Define the set of cosets of I in the additive group .

Is this definition well defined?

A 
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 ?
57
Q

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)

A

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.

58
Q

Define the Factor Ring of R by I.

A 
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.
59
Q

State the Fundamental Isomorphism Theorem

A

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

60
Q

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

A

(R/I)/(J/I) ≃ R/J

61
Q

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

A

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

62
Q

Define Maximal

A

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.

62
Q

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

A

the factor ring R/I is a field.

64
Q

Define a prime proper ideal

A

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

65
Q

POLYNOMIAL RINGS AND FACTORISATION

⚢ + ♂ = ?

A

GREAT FUN

66
Q

thm 5.1. State the division theorem for polynomials.

what are the quotient and remainder

A

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.

67
Q

Define a Root (of f)

A

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

68
Q

Corollary 5.4. ??

roots

A

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.

69
Q

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

A

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.

70
Q

Describe the Division algorithm

A

-

71
Q

Corollary 5.6.

gcd

A

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:

72
Q

Corollary 5.8.

A

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

73
Q

Define Irreducible

A

An element r ∈ R is irreducible if r is not invertible and

if, whenever r = st either s or t is invertible.

74
Q

Define Associated

A

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.

75
Q

Define a Unique Factorisation Domain

A

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.

76
Q

Define a Principal Ideal Domain

A

A principal ideal domain, is a commutative

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

77
Q

Lemma 6.2.

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

A

is a maximal ideal of K[X].

78
Q

Thm 6.3. State Kronecker’s Theorem

A

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

79
Q

Define Algebraically Closed

A

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

Algebraic Structures 2 Billy Cole (1 decks)

Brainscape helps you reach your goals faster, through stronger study habits.
© 2024 Bold Learning Solutions. Terms and Conditions