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 | i ∈ I, j ∈ J }.
   Prove that IJ ⊆ I ∩ J.
   The ideal I + J is the ideal generated by I ∪ J.
   Let I be the ideal < 4 > in Z and let J = < 6 > . Describe the ideals IJ, I ∩ J and I + J.

   Solution to question 1

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

   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 Z_mn and Z_m × Z_n are isomorphic both as groups and as rings.

   Solution to question 3

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

   Solution to question 4

5. Which of the maps from C to C given by the following are ring homomorphisms.
   x + yi ↦ x    x + yi ↦ x - yi    x + yi ↦ |x + iy|    x + yi ↦ y + 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 Z_n by U_n.
   For various values of n (as far as you can!) identify the groups U_n 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 Z_m onto Z_n ?

   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