Podcast
Questions and Answers
For a ring R and an element r ∈ R, which option correctly states that r is not a unit in R?
For a ring R and an element r ∈ R, which option correctly states that r is not a unit in R?
- R = Z6 and r = 6 (correct)
- R = M2 (R) and r = ( 01 10 )
- R = R[x] and r = x
- R is any ring, s, t ∈ R are units, and r = st
In the context of rings and elements, when is an element considered a unit?
In the context of rings and elements, when is an element considered a unit?
- When it is a prime number
- When it is the additive identity
- When it has an inverse in the ring (correct)
- When it commutes with all other elements
What is the definition of an injective homomorphism between two groups?
What is the definition of an injective homomorphism between two groups?
- A homomorphism that sends the identity to the identity element
- A homomorphism that has a non-trivial kernel
- A homomorphism that preserves the order of elements
- A homomorphism that collapses distinct elements to the same element (correct)
If G ∼ H, what does this imply about the groups G and H?
If G ∼ H, what does this imply about the groups G and H?
In modular arithmetic, what does it mean for x ≡ y mod n?
In modular arithmetic, what does it mean for x ≡ y mod n?
What is the kernel of a homomorphism in group theory?
What is the kernel of a homomorphism in group theory?
Is the set R = {a + b√2 | a, b ∈ Z} a field?
Is the set R = {a + b√2 | a, b ∈ Z} a field?
Which of the following statements about the set R = {a + b√2 | a, b ∈ Z} is FALSE?
Which of the following statements about the set R = {a + b√2 | a, b ∈ Z} is FALSE?
If H ∪ gH is a subgroup of G, what can be said about the order of g squared (g²)?
If H ∪ gH is a subgroup of G, what can be said about the order of g squared (g²)?
Which of the following best describes an equivalence relation?
Which of the following best describes an equivalence relation?
If there exists a ring homomorphism ϕ: R → Z₂, what can be concluded about R?
If there exists a ring homomorphism ϕ: R → Z₂, what can be concluded about R?
Why is the set R = {a + b√2 | a, b ∈ Z} not a field?
Why is the set R = {a + b√2 | a, b ∈ Z} not a field?
In which option is G not cyclic?
In which option is G not cyclic?
Which option does not define a group homomorphism?
Which option does not define a group homomorphism?
Which option has G not isomorphic to H?
Which option has G not isomorphic to H?
In which option does ∼ define an equivalence relation?
In which option does ∼ define an equivalence relation?
