Problem Set 1. Due Tuesday, January 16.
Chapter 1, Exercises 1, 2, 4, 5, 7, 8, 10, 12.
In Exercise 5, ignore the question about the converse in (ii).
Problem Set 2. Due Tuesday, January 30. Extended to
Tuesday, February 6.
Chapter 2, Exercises 2, 4, 5, 6, 7, 9, 10, and
Problem A: Let f(x) be a polynomial in one variable with integer coefficients.
Suppose there are finitely many primes p_{1}, ...,
p_{k} such that for each integer n, f(n) is divisible
by p_{i} for some i between 1 and k. Use the Chinese Remainder
Theorem to show that there exists an i between 1 and k, such that f(n)
is divisible by the same prime p_{i} for each integer n.
Problem Set 3. Due Tuesday, February 13.
Chapter 2, Exercises 1, 3, 12. In Exercise 1, prove the converse as well; the tensor product
is 0 if and only if m and n are relatively prime.
Chapter 6, Exercises 1, 2, 3, 4. In Exercise 4, answer only the first question
(about the Noetherian property).
Problem Set 4. Due Tuesday, March 6.
Problem Set 5. Due Tuesday, March 20.
Solution to Problem 1(a) from Problem Set 5.
Problem Set 6. Due Thursday, April 5
Midterm 1 will cover the following topics from Chapters 1, 2 and 6 in the text.
Chapter 1: Zorn's Lemma, rings and ideals, the nilradical and the Jacobson radical, the radical of an ideal, the Chinese Remainder Theorem.
Chapter 2: Submodules and quotient modules, direct sums and products of
modules, free modules, exact sequences, Nakayama's Lemma and its variants,
operations on modules (Hom and tensor product), exactness properties of tensor products, flat modules.
Chapter 6: Artinean and Noetherian rings and modules, connection
between the Noetherian and finitely generated modules, composition
series and modules of finite length, statement of the Hilbert Basis
Theorem.
Midterm 1 problems and solutions
Midterm 2 will cover the following topics: the Hilbert Basis theorem from Chapter 7, Grobner bases, rings of fractions from Chapter 3, integral ring extensions from Chapter 5. Khovanskii bases (SAGBI) and Noether Normalization will not be covered.
Midterm 2 problems and solutions