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 the integers, the set of all multiples of 6 is an ideal.

    Circle one:   True   False

  2. In the integers, the ideal (12, 18) equals (d) for one positive integer d. What is d?

    Answer: ______________

  3. Which of these subsets of the integers is an ideal?

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

  4. True or false: in the ring of integers Z, the set of all multiples of 6 is an ideal.

    Circle one:   True   False

  5. In the integers, for which positive n is the ideal (n) maximal?

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

  6. Let f: R -> S be a ring homomorphism. The kernel of f, meaning everything in R that f sends to 0, is always...

    • An ideal of R
    • A subfield of R
    • An ideal of S
    • Just a plain subset with no extra structure

  7. Which of these subsets of the ring of integers Z is an ideal?

    • The even integers
    • The odd integers
    • The positive integers
    • The set {0, 1, -1}

  8. Let f map R to S be a ring homomorphism. What is the kernel of f?

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

  9. What is the quotient ring Q[x]/(x^2 + 1)?

    • A field in which x acts like a square root of -1
    • A ring that has zero divisors
    • The ring of integers Z
    • The zero ring, with only one element

  10. What is the quotient ring Q[x] divided by (x squared plus 1)?

    • The integers
    • A field where x acts like a square root of minus 1
    • A ring with zero divisors
LightMySky · lightmysky.comW1-mt_uQBA9E1JIP-s1

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 is closed under subtraction and absorbs multiplication by any integer.
  2. 6 · The ideal generated by two integers is generated by their greatest common divisor.
  3. The even integers · Only the evens pass both the subtraction test and the absorption test.
  4. True · Multiples of 6 are closed under subtraction, and any integer times a multiple of 6 is still a multiple of 6, so the set is an ideal.
  5. Exactly when n is prime · Z divided by (n) is a field exactly when n is prime.
  6. An ideal of R · The kernel lives in R and passes both ideal tests: differences stay in the kernel, and multiplying a kernel element by anything in R stays in the kernel.
  7. The even integers · The even integers pass both ideal tests: subtracting two evens gives an even, and multiplying an even by any integer stays even.
  8. An ideal of the source ring R · The kernel lives in R and passes both ideal tests.
  9. A field in which x acts like a square root of -1 · Quotienting by (x^2 + 1) forces x^2 = -1, and because x^2 + 1 is irreducible over Q, the quotient is a field, namely Q(i).
  10. A field where x acts like a square root of minus 1 · The relation x squared equals minus 1 holds, and irreducibility makes the quotient a field.
Worksheet · LightMySky