3) Isomorphisms, Homomorphisms and Ideals Flashcards

1
Q

What does it mean for two rings to be isomorphic

A
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2
Q

What are the properties of an isomorphism

A
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3
Q

Describe the proof that an element r in a ring R is invertible if and only if its image θ(r) under a ring homomorphism θ:R→S is invertible in S

A
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4
Q

Describe the proof that an element r ∈ R is nilpotent iff θ(r) is nilpotent

A
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5
Q

What is the condition regarding the characteristic of rings for an isomorphism to exist between them

A

For an isomorphism to exist between two rings, they must have the same characteristic

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6
Q

If there is a homomorphism between two rings do they have the same characteristic

A

No, eg. θ : Z → Zn defined by θ(a) = [a]n.

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7
Q

Describe the proof that if two rings have an isomorphism between them then they must have the same characteristic

A
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8
Q

What does it mean if you have a homophorism between two rings

A
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9
Q

Give an example of a mapping that is a homomorphism but not a isomorphism

A

𝜙 : 𝑍 → 𝑍𝑛
This mapping is surjective but not injective, so it is a homomorphism but not an isomorphism.

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10
Q

Give an example of a homomorphism that is not surjective

A
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11
Q

What are the properties of an homophorism

A
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12
Q

Describe the proof that if there is a homophorism between R and S then the image of θ is a subring of S

A
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13
Q

What is an embedding or monomorphism

A

An injective homomorphism

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14
Q

What are the properties of composition with respect to ring homomorphisms and embeddings

A
  • Homomorphisms Composition: The composition of two ring homomorphisms is a homomorphism
  • Embeddings Composition: Composing two embeddings (injective homomorphisms) results in an embedding
  • Embedding via Composition: If the composition of two homomorphisms is an embedding, the initial homomorphism must also be an embedding
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15
Q

Describe the proof that the composition of two ring homomorphisms is homomorphism

A
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16
Q

Describe the proof that the initial homomorphism is embedding if composition is embedding

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17
Q

Give an example to show that that if θ : R −→ S and β : S −→ T are homomorphisms such that the composition β ◦ θ : R −→ T is an embedding then it need not be the
case that β is an embedding

18
Q

What is the kernel of a homomorphism

A

ker(θ) = {r ∈ R | θ(r) = 0}

19
Q

What determines the injectivity of a ring homomorphism

A

Iff ker(θ) = {0}

20
Q

Describe the proof that a homomorphism is injective iff ker(θ) = {0}

21
Q

What is an automorphism

A

An isomorphism from the ring to itself

22
Q

What are the key properties of the kernel of a ring homomorphism

23
Q

What is an ideal of a ring

24
Q

What defines a principal ideal generated by an element a

A

If a ∈ R then {r1as1 + · · · + rnasn | n ≥ 1, ri, si ∈ R} is an ideal which contains a and is the smallest ideal of R containing a
It is denoted as < a >

25
If a ring is commutative how can it's principal ideal be defined
26
What is a principle ideal
An ideal in a ring that can be generated by a single element of the ring
27
What is the trivial ideal and proper ideal
Trivial Ideal : < 0 > = {0} Proper Ideal: Anything not this -> < 1 > = R
28
Let R = Z[X] and let I = < 2, X > be the ideal of Z[X] generated by {2, X}. Show that I is not a principal ideal
29
What are the following ideal notations equivalent to - * < a , b> * < a > + < b > * < a > ∩ < b > * < a > < b >
< a , b> = < gcd(a,b) > < a > + < b > = < a , b > < a > ∩ < b > = < lcm(a,b) > < a > < b > = < ab >
30
When is a proper ideal a subring
Any proper ideal is not a subring, only if I = R then I is a subring
31
What is a right ideal
A right ideal is defined as for a general ideal but with the third condition replaced by the weaker condition: a ∈ I and r ∈ R implies ar ∈ I (and left ideals are defined similarly)
32
Give an example of a subset of a ring that does not form an ideal of the ring
Take R = Z and H = N ∪ {0}. H is not an ideal because 1 ∈ H but −1 × 1 ∈/ H.
33
What condition makes a commutative ring a field in terms of its ideals
Iff the only ideals of R are {0} and R
34
Describe the proof that a commutative ring R is a field iff the only ideals of R are {0} and R
Field to Ideals: In a field R, any non-zero a in a non-zero ideal I implies 1∈I, making I=R. Hence, only ideals are {0} and R. Ideals to Field: If {0} and R are the only ideals, for any non-zero r∈R, ⟨r⟩=R implies r is invertible since there is some elememt x such that rx = 1, making R a field
35
How is the ideal generated by a subset A of a commutative ring R defined
{a1r1 + · · · + anrn | n ≥ 1, ai ∈ A, ri ∈ R}
36
What is the nature of the kernel of a ring homomorphism
If θ : R → S is a homomorphism of rings, then ker(θ) is an ideal of R
37
Describe the proof that if θ : R → S is a homomorphism of rings, then ker(θ) is an ideal of R
* We need to show that ker(θ) satisfies the three conditions for an ideal. * Firstly 0 ∈ ker(θ). Let r, r' ∈ ker(θ). Then θ(r + r') = θ(r) + θ(r') = 0 + 0 = 0 and so r + r' ∈ ker(θ) * Let s ∈ R. Then θ(sr) = θ(s)θ(r) = θ(s) × 0 = 0 and similarly θ(rs) = 0 * Therefore sr, rs ∈ ker(θ)
38
What can be said about a ring homomorphism when R is a field
If θ : R → S is a homomorphism of rings and R is a field then θ is a monomorphism
39
What are the properties of sums and intersections of ideals in a ring
40
Describe the proof that if I, J are ideals of a ring R then IJ ⊆ I ∩ J
The typical element of IJ is a sum of terms of the form ab with a ∈ I and b ∈ J. Note that any such element ab is both in I and in J, hence is in I ∩ J. Hence any sum of such elements is in I ∩ J. That is, IJ ⊆ I ∩ J
41
What is a product of ideals