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Math 512 Problem Set 6

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Math 512 Problem Set 6
Exercise 1. Let R be a ring in which every maximal ideal is of the form cR
for some c ∈ R satisfying c
2 = c. Show that R is Noetherian (Hint: Show
that every prime ideal is maximal).
Exercise 2. Let (R,M) be a Noetherian local ring. Suppose that M/M2
is
generated by the set {a1+M2
, . . . , an+M2}. Show that M = a1R+· · ·+anR.
Exercise 3. Let R ⊂ S be an integral extension, and suppose that R and S
are both integral domains. Show that R is a field if and only if S is a field.
Exercise 4. Show that if R ⊂ S is an integral extension, then S[x1, . . . , xn]
is integral over R[x1, . . . , xn].
Exercise 5. Let R be an integral domain with fractional field k. Show that
if R is integrally closed and t is transcendental over k, then R[t] is integrally
closed.
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