Rings and Fields

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(Exercises 3)
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(Exercises 5)

Exercises 4

  1. Prove that the intersection of two ideals of a ring is an ideal.
    If I and J are ideals of a commutative ring with identity, let IJ be the ideal generated by the set {ij | iI, jJ }.
    Prove that IJIJ.
    The ideal I + J is the ideal generated by IJ.
    Let I be the ideal < 4 > in Z and let J = < 6 > . Describe the ideals IJ, IJ and I + J.

    Solution to question 1

  2. Prove that the ring Zn is a principal ideal domain for any n. How would you determine the number of ideals of Zn ?

    Solution to question 2

  3. Prove that 2Z and 3Z are isomorphic as abelian groups but not as rings (under, of course, the usual addition and multiplication).
    The addititive groups Z6 and Z2 × Z3 are isomorphic as groups. Show that they are also isomorphic as rings.
    [Hint: A suitable group isomorphism maps 1 ∈ Z6 to (1, 1) ∈ Z2 × Z3 .]
    More generally, show that if m, n are coprime integers then Zmn and Zm × Zn are isomorphic both as groups and as rings.

    Solution to question 3

  4. Prove that the map from Z12 to Z4 given by nn mod 4 for nZ12 is a ring homomorphism. What is its kernel?
    Is the map from Z14 to Z4 given by nn mod 4 for nZ14 a ring homomorphism?

    Solution to question 4

  5. Which of the maps from C to C given by the following are ring homomorphisms.
    x + yix    x + yix - yi    x + yi ↦ |x + iy|    x + yiy + xi

    Solution to question 5

  6. Show that the units (multiplicatively invertible elements) in any ring form a group under multiplication.
    Denote the group of units of Zn by Un.
    For various values of n (as far as you can!) identify the groups Un as products of cyclic groups. Can you spot any pattern? Question 3 above should be helpful.

    Solution to question 6

  7. Describe all the ring homomorphisms from Z12 onto Z4 .
    Describe all the ring homomorphisms from Z12 to Z5 .
    For which values of m, n can one find a ring homomorphism from Zm onto Zn ?

    Solution to question 7

  8. Look at possible addition and multiplication tables and prove that up to isomorphism there are two rings of order 2.
    If the additive group of a ring is cyclic, generated (say) by an element a, prove that the multiplication is determined once you know a.a . Hence determine how many different rings of order 3 there are.

    Prove that there are three non-isomorphic rings of order 4 whose additive group is cyclic.

    If you have sufficient determination you can try and establish how many non-isomorphic rings there are whose additive group is the Klein 4-group Z2 × Z2 .

    Solution to question 8

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JOC/EFR 2004