Assignment 2, Due 10/09 Before Class Begins

ASSIGNMENT 2, DUE 10/09 BEFORE CLASS BEGINS

RANKEYA DATTA

You may talk to each other while attempting the assignment, but you must write up your own solutions, preferably on TeX. You must also list the names of all your collaborators. You may not use resources such as MathStackExchange and MathOverflow. Please put some effort into solving the assignments since they will often develop important theory I will use in class. The assignments have to be uploaded on Gradescope. Please note that Gradescope will not accept late submissions. All rings are commutative. Problem 1. In this problem you will construct a non-noetherian ring. Consider the group G := Z ⊕ Z, with a total ordering as follows: (a1, b1) < (a2, b2) if and only if either a1 < a2 or a1 = a2 and b1 < b2. Here total ordering means that any two elements of G are comparable. Define a function ν : C(X,Y ) − {0} → G P α β as follows: for a polynomial f = (α,β) cαβX Y ∈ C[X,Y ] − {0},

ν(f) := min{(α, β): cαβ 6= 0}. For an arbitrary element f/g ∈ C(X,Y ) − {0}, where f, g are polynomials, ν(f/g) := ν(f) − ν(g). (1) Show that ν is a well-deﬁned group homomorphism. (2) Show that for all x, y ∈ C(X,Y ) − {0} such that x + y 6= 0, ν(x + y) ≥ min{ν(x), ν(y)}. (3) Show that Rν := {x ∈ C(X,Y )−{0} : ν(x) ≥ (0, 0)}∪{0} is a subring of C(X,Y ) containing C[X,Y ]. (4) Show that Rν is a local ring with maximal ideal mν = {x ∈ C(X,Y ) − {0} : ν(x) > (0, 0)} ∪ {0}. (5) Show that for all x, y ∈ Rν, x|y of y|x. (6) Show that if I,J are ideals of Rν, then I ⊆ J of J ⊆ I. (7) Show that for all x, y ∈ Rν − {0}, (x) = (y) if and only if ν(x) = ν(y). (8) Show that Rν is not noetherian by ﬁnding an ascending chain of ideals that does not stabilize.

ν is an example of a valuation and Rν is an example of a valuation ring.

Problem√ 2. Let R be a noetherian ring. Prove that for any ideal I of R, there exists n > 0 such that ( I)n ⊆ I. Give an example of a non-noetherian ring R and an ideal I of R for which this property fails. √ Problem 3. Let R be a ring. If I is a finitely generated ideal of R, is I finitely generated? Give complete justification. Problem 4. Let R be a ring and I be an ideal of R. Prove that if x ∈ R such that (I, x) and (I :(x)) are both finitely generated, then I is finitely generated. Problem 5. Prove that a ring R is noetherian if and only if all prime ideals of R are finitely generated via the following steps for the harder of the two implications: 1 (1) Let Σ be the collection of ideals of R that are not finitely generated. If Σ 6= ∅, show using Zorn’s Lemma that Σ has a maximal element. (2) Show any maximal element of Σ must be prime. Hint: Use Problem 4. Show that if all maximal ideals of a ring R are finitely generated, then R need not be noetherian. Hint: Use Problem 1. Problem 6. A topological space X is noetherian if every open subset of X is quasi-compact.

(1) Show that if R is a ring, and f ∈ R, then the set D(f) = Spec(R) − V(f) is quasi-compact. (2) Show that Spec(R) is quasi-compact, for any ring R. (3) Show that a ﬁnite union of quasi-compact subsets of a topological space is quasi-compact. Use this to show that if R is a noetherian ring, then Spec(R) is a noetherian topological space. (4) With complete proof, give an example of a ring R such that Spec(R) is a noetherian topological space but R is not noetherian.

Problem 7. Let p be a prime ideal of a noetherian ring R. Let q : R → Rp be the localization map. The n-th symbolic power of p, denoted p(n), is deﬁned to be the ideal −1 n q (p Rp). (1) Show that p(n) is a p-primary ideal of R. (2) Show that p(n) is the unique p-primary component in a primary decomposition of pn. (3) Show that pn is p-primary if and only if pn = p(n). (4) Show that for all maximal ideals m of R, m(n) = mn. 3 (5) Consider the ring R = C[x, y, z]/(x − yz). Consider the prime ideal p = (x, y)/(x3 − yz) of R. Show that y ∈ p(2) but y∈ / p2. Hence, symbolic and ordinary powers need not coincide. This also shows that an ideal whose radical is prime need not be primary.

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