Commutative Algebra, Problem Set 11
Denition: Let P A be a prime ideal. The height of P is the Krull dimension of the
local ring AP .
(1) Let A be noetherian local ring. Suppose that there exists a principal prime ideal in
A of height at least 1. Prove th
Commutative Algebra, Problem Set 7
(1) Show that a UFD is integrally closed.
(2) Suppose that A B is an integral extension of rings and that B is a nitely generated
algebra over A. Show that B is a nitely generated A-module.
(3) Let A B be an integral ext
Commutative Algebra, Problem Set 10
(1) Show that any nitely generated ideal in a valuation ring is principal.
(2) Let A be a DVR with quotient eld k. Show that A is a maximal propert subring of
k.
(3) Show that a valuation ring which is not a eld, is noe
Commutative Algebra, Problem Set 6
(1) Let (A, m) (B, n) be local rings. Show that m n m = n A. If this is the case,
we say B dominates A.
(2) Let A be an integral domain with quotient eld k. We say that A is a valuation ring
of k if for every 0 = x k, ei
Commutative Algebra, Problem Set 9
(1) Let I be an irreducible ideal in a ring A. Show that the following are equivalent.
(a) I is primary.
(b) For every multiplicative set S in A, (S 1 I)c = (I : x) for some x S.
(c) The sequence (I : xn ) stabilizes for
(1)
(2)
(3)
(4)
(5)
(6)
Commutative Algebra, Problem Set 8
Let f : A B be an integral homomorphism of rings (f need not be injective). This
means that B is integral over f (A). Show that the induced map of topological spaces
f : Spec A Spec B is closed.
L
Commutative Algebra, Problem Set 5
(1) Let R be a ring such that for every maximal ideal P R the local ring RP is
noetherian. Further assume that for every a R, there are only nitely many maximal
ideals containing x. Show that R is noetherian.
(2) Let A b
Commutative Algebra, Problem Set 4
(1) Let S A be multiplicative. Show that the canonical map A AS is an isomorphism
if and only if S consists of units.
(2) Let a A and S = cfw_1, a, a2 , . . .. Let A[X] be the polynomial ring in one variable
over A. Show
Commutative Algebra, Problem Set 2
(1) Show that a ring R is noetherian every ideal of R is nitely generated every
prime ideal of R is nitely generated.
(2) Let R be a ring and let M be a module. Assume that M has a composition series.
Show that the lengt
HW-6: 4th November 2013
Throughout R denotes a commutative ring with identity and k a eld.
1. (5 points) Prove the sublemma from 21st October 2013.
2. (10 points) Show that the following are equivalent:
1. R is Artinian;
2. R is Noetherian and zero-dimens
HW-7: 18th November 2013
Throughout R denotes a commutative ring with identity and k a eld.
1. (5 points) Let k be a eld of positive characteristic p and R a k-algebra. Show that the map F : R
R, r rp is a ring morphism. (This is called the Frobenius end
Commutative Algebra, Problem Set 1
(1) Let R be a ring, and I an ideal consisting of nilpotent elements. Suppose that a R
maps to a unit in R/I. Then show that a is a unit of R.
(2) Show that if R is a UFD then R[X] is also a UFD.
(3) Show that prime elem
Throughout R denotes a commutative ring with identity and k a eld.
HW-2: 2nd September 2013
1. (5 points) Fill in the details of the proof of NAK from Serre, Local Algebra, Chapter I, Section 1. More
precisely, justify why M has a quotient of the form A/m
Throughout R denotes a commutative ring with identity and k a eld.
HW-5: 21st October 2013
1. (5 points) Atiyah-MacDonald Chapter 4 Exercise 2. (Resubmit your solution even if you submitted it
last time.)
2. (5 points) Atiyah-MacDonald Chapter 4 Exercise
Commutative Algebra, Problem Set 3
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(12)
(13)
(14)
If a ring R is artinian, then is the polynomial ring R[X] also artininan?
If R[X] is noetherian, then is R noetherian?
Is Q/Z artinian as a Z-module?
Let (R, m
Throughout R denotes a commutative ring with identity and k a eld.
HW-4: 7th October 2013
1. (5 points) Using the prime avoidance lemma (Lemma 3.3, Eisenbud) (or otherwise), show that if (R, m)
is a Noetherian local ring and M is a nitely generated R-modu
Throughout R denotes a commutative ring with identity and k a eld.
HW-3: 17nd September 2013
(17th is a Tuesday; submit this in class in the afternoon.)
1. (5 points) Atiyah-MacDonald Chapter 3 Exercise 1.
2. (5 points) Atiyah-MacDonald Chapter 3 Exercise
Throughout R denotes a commutative ring with identity and k a eld.
HW-1: 19th August 2013
1. (5 points) Let R be a domain and f R[X]. Then f is a unit if and only if deg f = 0 and f is a unit
in R.
2. (5 points) Let A be a subset of R. Show that the small