189-235A: Basic Algebra I Assignment 9

Due: Wednesday, November 17

1. Show that the intersection of two prime ideals need not be a prime ideal.

2. Show that a ring R is an integral domain if and only if {0} is a prime ideal.

3. If P is a prime ideal in a commutative ring R, is the ideal P × P a prime ideal of R × R?

4. Find an ideal of Z × Z that is prime but not maximal.

5. Let R be a commutative ring, let F [x] be the ring of polynomials with coefficients in a field F, and let f : R ⟶ F [x] be a ring homomorphism. Show that the kernel of f is a prime ideal of R.

6. Let F be a field, and define a binary composition law on G = F − {1} by the rule

a ∗ b = a + b − ab.

Show that G, with this operation, is a group. (In particular, write down the neutral element for ∗, and give a formula for the inverse of a ∈ G.

7. List all the elements of order 3 in S_{3}. How many are there?

8. List all the elements of order 6 in S_{5}. How many such elements are there?

9. Given an example of non-abelian groups of order 12 and 30.

10. Extra credit question. Suppose that G is a group in which x^{2} = 1, for all x ∈ G. Show that G is abelian. Give an example of a non-abelian group G of order 27 in which x^{3} = 1 for all x ∈ G.