Mathematics 667
Homework (due Mar. 11)
A. Hulpke
20) A (noetherian, but that is just for the definition) ring is called local if it has exactly one maximal
ideal.
a) Show that a ring A is local if and only if the non-units in A form an ideal.
b) If A is a local ring with maximal ideal J = Jac(A) ⊲ A, show that ∩J n = 0. (Hint: Consider the
intersection as module and use the Nakayama Lemma)
c) Let A be a noetherian ring, P ⊲ A a prime ideal, and S = A − P. Show that S −1 A is a local ring.
21) Let P ⊲ A a prime ideal and S = A − P. Let B = S −1 A and consider the natural map f ∶ A → B.
a) Let Q ⊲ A be a prime ideal. When is Q e nonzero? When is Q e a prime ideal?
b) Let Q ⊲ B a prime ideal. When is Q c a prime ideal of A?
22) Let J ⊲ A and let S = 1 + J = {1 + j ∣ j ∈ J}. Show that S −1 J is contained in the Jacobson radical
of S −1 A.
23) Let A be a ring and S, T ⊂ A both multiplicatively closed. Let U be the image of T in S −1 A.
Show that (ST)−1 A and U −1 (S −1 A) are isomorphic as rings.
24) Let M ⊲ k[x1 , . . . , x n ] be a monomial ideal. Give creteria for M being a prime ideal, respectively being equal to its own radical.
25) A multiplicatively closed subset S ⊂ A is called saturated if
x y ∈ S ⇔ x ∈ S and y ∈ S.
a) Show that S is saturated if and only if A − S is a union of prime ideals.
b) Show that for each multiplicatively closed set S ⊂ A, there is a unique smallest multiplicatively
closed set S ⊂ S̄ ⊂ A that is saturated. (One has that S̄ = A − ⋃P∩S=∅ P, where the P ⊲ A are prime
ideals. S̄ is called the saturation of S.