**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.