Unit Element in a Ring Quiz
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Questions and Answers

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?

    <p>They are isomorphic groups</p> Signup and view all the answers

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

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

    What is the kernel of a homomorphism in group theory?

    <p>The set of all elements mapped to the identity element</p> Signup and view all the answers

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

    <p>No</p> Signup and view all the answers

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

    <p>R and Z × Z are isomorphic rings</p> 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)?

    <p>ord(g²) divides |H|</p> Signup and view all the answers

    Which of the following best describes an equivalence relation?

    <p>A relation that is reflexive, symmetric, and transitive</p> Signup and view all the answers

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

    <p>R is not a field</p> Signup and view all the answers

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

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

    In which option is G not cyclic?

    <p>G is a nonabelian group</p> Signup and view all the answers

    Which option does not define a group homomorphism?

    <p>G = S3 and H = S4</p> Signup and view all the answers

    Which option has G not isomorphic to H?

    <p>G = S3 and H = D3</p> Signup and view all the answers

    In which option does ∼ define an equivalence relation?

    <p>S = R, x ∼ y if |x − y| &lt; 1</p> Signup and view all the answers

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