3) Isomorphisms, Homomorphisms and Ideals Flashcards
What does it mean for two rings to be isomorphic
What are the properties of an isomorphism
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
Describe the proof that an element r ∈ R is nilpotent iff θ(r) is nilpotent
What is the condition regarding the characteristic of rings for an isomorphism to exist between them
For an isomorphism to exist between two rings, they must have the same characteristic
If there is a homomorphism between two rings do they have the same characteristic
No, eg. θ : Z → Zn defined by θ(a) = [a]n.
Describe the proof that if two rings have an isomorphism between them then they must have the same characteristic
What does it mean if you have a homophorism between two rings
Give an example of a mapping that is a homomorphism but not a isomorphism
𝜙 : 𝑍 → 𝑍𝑛
This mapping is surjective but not injective, so it is a homomorphism but not an isomorphism.
Give an example of a homomorphism that is not surjective
What are the properties of an homophorism
Describe the proof that if there is a homophorism between R and S then the image of θ is a subring of S
What is an embedding or monomorphism
An injective homomorphism
What are the properties of composition with respect to ring homomorphisms and embeddings
- 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
Describe the proof that the composition of two ring homomorphisms is homomorphism
Describe the proof that the initial homomorphism is embedding if composition is embedding
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
What is the kernel of a homomorphism
ker(θ) = {r ∈ R | θ(r) = 0}
What determines the injectivity of a ring homomorphism
Iff ker(θ) = {0}
Describe the proof that a homomorphism is injective iff ker(θ) = {0}
What is an automorphism
An isomorphism from the ring to itself
What are the key properties of the kernel of a ring homomorphism
What is an ideal of a ring
What defines a principal ideal generated by an element 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 >
If a ring is commutative how can it’s principal ideal be defined
What is a principle ideal
An ideal in a ring that can be generated by a single element of the ring
What is the trivial ideal and proper ideal
Trivial Ideal : < 0 > = {0}
Proper Ideal: Anything not this -> < 1 > = R
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
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 >
When is a proper ideal a subring
Any proper ideal is not a subring, only if
I = R then I is a subring
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)
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.
What condition makes a commutative ring a field in terms of its ideals
Iff the only ideals of R are {0} and R
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
How is the ideal generated by a subset A of a commutative ring R defined
{a1r1 + · · · + anrn | n ≥ 1, ai ∈ A, ri ∈ R}
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
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)
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
What are the properties of sums and intersections of ideals in a ring
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
What is a product of ideals
