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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).