Ideals and Quotient Rings · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Ideals and what quotients do

Mathematics · Abstract Algebra · ages 22-23
Name ______________________   Date ____________

  1. In Z, the set of all multiples of 6 is an ideal.

    Circle one:   True   False

  2. In Z, (12, 18) = (d) for a positive generator d. What is d?

    Answer: ______________

  3. Which subset of Z is an ideal?

    • The positive integers
    • The odd integers
    • The even integers

  4. The empty subset is an ideal because it has no subtraction or multiplication counterexample.

    Circle one:   True   False

  5. For which positive n is the integer ideal (n) maximal?

    • Exactly when n is even
    • Exactly when n is a perfect square
    • Exactly when n is prime

  6. An ideal I of a commutative ring R with identity contains 1. What follows?

    • I contains only the elements 0 and 1
    • I = R
    • I must be a proper maximal ideal
    • I contains no other invertible element

  7. The even integers form an ideal of Z. Why do they not form an ideal of Q?

    • Multiplication by 1/2 leaves the subset
    • Subtracting two even integers produces an odd integer
    • The rational ring has no multiplicative identity
    • The subset contains no additive identity

  8. Let f: R -> S be a ring homomorphism between commutative rings with identity. What is its kernel?

    • An ideal of R
    • A subfield of the source ring R
    • An ideal of the target ring S

  9. In Q[x]/(x^2 - 4), which pair of nonzero classes has zero product?

    • [x - 1] and [x + 1]
    • [x] and [x]
    • [1] and [x + 2]
    • [x - 2] and [x + 2]

  10. Which describes Q[x]/(x^2 + 1), with u the class of x?

    • An integer ring where u^2 = 1
    • A field with u^2 = -1
    • A ring with nonzero zero divisors
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Answer key

For grown-ups. Fold this page away before handing over the rest.

Ideals and what quotients do W1-mt_uQBA9E1JIP-s1

  1. True · It contains 0, closes under differences and absorbs every integer multiplier.
  2. 6 · Every combination is divisible by 6, and 18 - 12 produces 6.
  3. The even integers · The even integers contain 0 and satisfy both closure conditions.
  4. False · An ideal must contain 0, which the empty subset does not.
  5. Exactly when n is prime · Prime n gives a field quotient; n = 1 is not proper and composite n gives zero divisors.
  6. I = R · Absorption puts every r = r times 1 into I.
  7. Multiplication by 1/2 leaves the subset · The ambient ring Q permits multiplier 1/2, and (1/2) times 2 is 1, not an even integer.
  8. An ideal of R · The inputs sent to 0 contain 0 and satisfy both ideal closure conditions in R.
  9. [x - 2] and [x + 2] · Their product is [x^2 - 4] = 0, and neither linear polynomial belongs to the degree-2 principal ideal.
  10. A field with u^2 = -1 · The quadratic has no rational root and is irreducible, so its quotient is a field.
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