Rings and Fields

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## 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.

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

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.

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?

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

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.

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 ?

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 .

SOLUTIONS TO WHOLE SET
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JOC/EFR 2004