Math 689 Commutative and Homological Algebra
Homework Assignment 1
Due Friday September 18
All rings are commutative with 1 6= 0.
1. Let A be a ring and let S be the set of all multiplicative subsets of A that do not
contain 0, ordered by inclusion. Prove that an element S of S is maximal in S if, and
only if, A − S is a minimal prime ideal (that is, a prime ideal that does not properly
contain any other prime ideal).
2. Let I and J be ideals of a ring A. Show that
√
√
√
√
IJ = I ∩ J = I ∩ J.
√
3. p
Let φ : A → B be a ring homomorphism, and I an ideal of B. Prove that φ−1 ( I) =
φ−1 (I).
4. Let n be a positive integer, n ≥ 2. Find nil(Z/(n)).
5. Let φ : A → B be a surjective ring homomorphism.
(a) Prove that φ(rad(A)) ⊂ rad(B).
(b) Give an example to show that the inclusion need not be an equality.
6. Let A be a local ring. Prove that A contains no idempotent elements other than 0
and 1. (An element a of A is idempotent if a2 = a.)
7. Let A be a local ring and M, N finitely generated A-modules. Prove that if M ⊗A N =
0, then M = 0 or N = 0.
8. Let A be a ring. Suppose that Am ∼
= An as A-modules for some positive integers m, n.
Prove that m = n. (Hint: Reduce this to the case of a field. Note that the statement is
false for noncommutative rings; a counterexample is in Hungerford, Algebra, exercise
IV.2.13.)
9. Let I1 , . . . , In be ideals of a ring A for which I1 ∩ · · · ∩ In = (0). Prove that if A/Ij is
Noetherian for each j, then A is Noetherian.
10. Let A, B, C be rings and suppose that φ : A → C, ψ : B → C are ring homomorphisms.
The fiber product A ×C B is defined to be the following subring of A × B:
A ×C B = {(a, b) ∈ A × B | φ(a) = ψ(b)}.
Prove that if A and B are Noetherian, and φ and ψ are surjective, then A ×C B is
Noetherian. (Hint: Use # 9.)