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Math 512 Problem Set 5 

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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.
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