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189-571B: Higher Algebra II

Assignment 1. Due: Wednesday, January 24.

All the questions are taken from Basic Algebra Chapter VIII, pages 443-445.

1. Question 1. (a)-(d).
(e) What if ${\bf R}$ is replaced by the ring ${\bf C}$ of complex numbers in this question?

2. Question 5.

3. Question 6.
This problem gives a prototypical instance of the very important principle whereby the maximal ideals in the ring of functions on a space $X$ are in natural bijection with the points of $X$. This simple principle is the soure of the tight connection between commutative ring theory and geometry.

4. Question 7.
This gives an instance where the principle described problem 3 fails! The space $X$ is not even very pathological; it just fails to be compact.

5. Question 9.

6. Question 10.

7. Question 13.

8. Question 16.
This question shows that the structure of ideals can be quite complicated, in even the most simple rings that are not PID's.

9. Question 17.

10. Question 20.