3) Isomorphisms, Homomorphisms and Ideals

Question 1

What does it mean for two rings to be isomorphic

Answer 1

Question 2

What are the properties of an isomorphism

Answer 2

Question 3

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

Answer 3

Question 4

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

Answer 4

Question 5

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

Answer 5

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

Question 6

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

Answer 6

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

Question 7

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

Answer 7

Question 8

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

Answer 8

Question 9

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

Answer 9

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

Question 10

Give an example of a homomorphism that is not surjective

Answer 10

Question 11

What are the properties of an homophorism

Answer 11

Question 12

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

Answer 12

Question 13

What is an embedding or monomorphism

Answer 13

An injective homomorphism

Question 14

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

Answer 14

• 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

Question 15

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

Answer 15

Question 16

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

Answer 16

Question 17

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

Answer 17

Question 18

What is the kernel of a homomorphism

Answer 18

ker(θ) = {r ∈ R | θ(r) = 0}

Question 19

What determines the injectivity of a ring homomorphism

Answer 19

Iff ker(θ) = {0}

Question 20

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

Answer 20

Question 21

What is an automorphism

Answer 21

An isomorphism from the ring to itself

Question 22

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

Answer 22

Question 23

What is an ideal of a ring

Answer 23

Question 24

What defines a principal ideal generated by an element a

Answer 24

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 >

Question 25

If a ring is commutative how can it’s principal ideal be defined

Answer 25

Question 26

What is a principle ideal

Answer 26

An ideal in a ring that can be generated by a single element of the ring

Question 27

What is the trivial ideal and proper ideal

Answer 27

Trivial Ideal : < 0 > = {0}
Proper Ideal: Anything not this -> < 1 > = R

Question 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

Answer 28

Question 29

What are the following ideal notations equivalent to -
* < a , b>
* < a > + < b >
* < a > ∩ < b >
* < a > < b >

Answer 29

< a , b> = < gcd(a,b) >
< a > + < b > = < a , b >
< a > ∩ < b > = < lcm(a,b) >
< a > < b > = < ab >

Question 30

When is a proper ideal a subring

Answer 30

Any proper ideal is not a subring, only if
I = R then I is a subring

Question 31

What is a right ideal

Answer 31

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)

Question 32

Give an example of a subset of a ring that does not form an ideal of the ring

Answer 32

Take R = Z and H = N ∪ {0}. H is not an ideal because 1 ∈ H but
−1 × 1 ∈/ H.

Question 33

What condition makes a commutative ring a field in terms of its ideals

Answer 33

Iff the only ideals of R are {0} and R

Question 34

Describe the proof that a commutative ring R is a field iff the only ideals of R are {0} and R

Answer 34

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

Question 35

How is the ideal generated by a subset A of a commutative ring R defined

Answer 35

{a1r1 + · · · + anrn | n ≥ 1, ai ∈ A, ri ∈ R}

Question 36

What is the nature of the kernel of a ring homomorphism

Answer 36

If θ : R → S is a homomorphism of rings, then ker(θ) is an ideal of R

Question 37

Describe the proof that if θ : R → S is a homomorphism of rings, then ker(θ) is an ideal of R

Answer 37

• 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)

Question 38

What can be said about a ring homomorphism when R is a field

Answer 38

If θ : R → S is a homomorphism of rings and R is a field then θ is a monomorphism

Question 39

What are the properties of sums and intersections of ideals in a ring

Answer 39

Question 40

Describe the proof that if I, J are ideals of a ring R then IJ ⊆ I ∩ J

Answer 40

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

Question 41

What is a product of ideals

Answer 41