Unit Element in a Ring Quiz

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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?

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?

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?

They are isomorphic groups Signup and view all the answers

In modular arithmetic, what does it mean for x ≡ y mod n?

$x$ and $y$ have the same remainder when divided by $n$ Signup and view all the answers

What is the kernel of a homomorphism in group theory?

The set of all elements mapped to the identity element Signup and view all the answers

Is the set R = {a + b√2 | a, b ∈ Z} a field?

No Signup and view all the answers

Which of the following statements about the set R = {a + b√2 | a, b ∈ Z} is FALSE?

R and Z × Z are isomorphic rings Signup and view all the answers

If H ∪ gH is a subgroup of G, what can be said about the order of g squared (g²)?

ord(g²) divides |H| Signup and view all the answers

Which of the following best describes an equivalence relation?

A relation that is reflexive, symmetric, and transitive Signup and view all the answers

If there exists a ring homomorphism ϕ: R → Z₂, what can be concluded about R?

R is not a field Signup and view all the answers

Why is the set R = {a + b√2 | a, b ∈ Z} not a field?

It doesn't have a multiplicative inverse for every non-zero element Signup and view all the answers

In which option is G not cyclic?

G is a nonabelian group Signup and view all the answers

Which option does not define a group homomorphism?

G = S3 and H = S4 Signup and view all the answers

Which option has G not isomorphic to H?

G = S3 and H = D3 Signup and view all the answers

In which option does ∼ define an equivalence relation?

S = R, x ∼ y if |x − y| < 1 Signup and view all the answers

Unit Element in a Ring Quiz

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