Math 689 Commutative and Homological Algebra
Homework Assignment 2
Due Friday October 9
All rings are commutative with 1 6= 0.
1. Let A, B be rings. Let S = {(1, 1), (1, 0)}, a multiplicative subset of A × B. Show
that (A × B)S ∼
= A.
2. Let A be a ring, and S a multiplicative √
subset of
√ A.
(a) Let I be an ideal of A. Show that IS = ( I)S .
(b) Show that nil(AS ) = (nil(A))AS .
3. Let A be a ring, and S a multiplicative subset of A. Show that if M is a projective
A-module, then MS is a projective AS -module.
√
4. Let k be a field. Show that the ideal I = (x2 , y) of k[x, y] is primary, and find I.
5. Let k be a field, and A = k[x, y, z]/(xy − z 2 ). Use an overline to denote an element of
2
2
2
2
A, i.e. x = x + (xy
√ − z ). Let I = (x, z) and J = I = (x , xz, z ), ideals of A. Show
that I is prime, J = I, and J is not primary.
6. Let A be the ring of all continuous real-valued functions on the unit interval [0, 1],
under pointwise addition and multiplication.
(a) For each c ∈ [0, 1], let Ic = {f ∈ A | f (c) = 0}. Show that Ic is a maximal ideal
of A.
(b) Show that each maximal ideal of A is of the form Ic for some c ∈ [0, 1], and
conclude that mSpec(A) is in one-to-one correspondence with the interval [0, 1].
7. Let A and B be rings. Show that Spec(A × B) can be identified (as a set) with the
˙
disjoint union Spec(A) ∪Spec(B).
8. Let f : A → B be a ring homomorphism, and af :Spec(B) →Spec(A) be the induced
map. Show that if I is an ideal of A, then (af )−1 (V (I)) = V (IB).