## Description

Math 512 Problem Set 5

Exercise 1. Let R be a PID, and I ⊂ R an ideal. Show that R/I is both

Noetherian and Artinian.

Exercise 2. Let R be Noetherian, and P ⊂ R a prime ideal. Show that RP

is Noetherian.

Exercise 3. Let R be an Artinian ring. Show that every prime ideal of R

is maximal.

Exercise 4. Let R be a ring, S ⊂ R a multiplicative set, and I ⊂ R an

ideal. Show that S

−1

(rad I) = rad(S

−1

I).

Exercise 5. Let R be Noetherian, and I, J ⊂ R ideals with J ⊂ rad I.

Show that there exists n ∈ N with J

n ⊂ I.

1