MATH32012
Two hours
THE UNIVERSITY OF MANCHESTER
COMMUTATIVE ALGEBRA
2013
Answer ALL FOUR questions in Section A (40 marks in total). Answer TWO of the THREE
questions in Section B (40 marks in total). If more than TWO questions from Section B are
attempt

MATH32012
SOLUTIONS
SECTION A
A1. All rings in this question are understood to be commutative.
(a) Give the denition of a noetherian ring.
Answer. A noetherian ring is a ring where every ideal is nitely generated.
(b) Give the denition of an irreducible e

MATH32012
Two hours
THE UNIVERSITY OF MANCHESTER
COMMUTATIVE ALGEBRA
18 May 2012
14:00 16:00
Answer ALL questions in Section A (40 marks) and TWO of the THREE questions in Section B
(20 marks each). The total number of marks on the paper is 80. A further

MATH32012
SOLUTIONS
Two hours
THE UNIVERSITY OF MANCHESTER
COMMUTATIVE ALGEBRA
18 May 2012
14:00 16:00
Answer ALL questions in Section A (40 marks) and TWO of the THREE questions in Section B
(20 marks each). The total number of marks on the paper is 80.

Assessed homework model solutions
[draft]
1
Assessed homework model solutions
HW1. [10 marks] Let I, J be ideals of Q[X1 , . . . , Xn ] such that I is a monomial ideal and
I J. Let
be a monomial ordering on M(X1 , . . . , Xn ). Carefully prove that I LT (

Commutative algebra coursework
HW1
Most student did this question well, some used a Grbner basis when it was not really required but still
o
proved the result which is ne. On the other hand some used the Grbner basis wrongly, in these questions
o
be caref

MATH32012
SOLUTIONS
Two hours
THE UNIVERSITY OF MANCHESTER
COMMUTATIVE ALGEBRA
31 May 2013
14:00 16:00
Answer ALL FOUR questions in Section A (40 marks in total). Answer TWO of the THREE
questions in Section B (40 marks in total). If more than TWO questio

Answers to exercises for 1
[revised v1 15/02/2014]
1
Answers to exercises for 1
E1.1 (Algebraic Structures 2 revision). Deduce from the axioms of a commutative ring that
0a = 0,
(1)a = a,
(a)b = a(b) = (ab)
for all a, b R.
Answer (not given in class). (i)

Answers to exercises for 5
[revised v2 29/03/2014]
1
Answers to exercises for 5
Remark: a mock test where students can practise nding Gr bner bases has been set up
o
online on Blackboard.
E5.1. Recall that Buchbergers algorithm with respect to Lex with X

Answers to exercises for 7
[revised v1 07/04/2014]
1
Answers to exercises for 7
E7.1. Prove the following, where I, J are ideals of a ring R:
(i) I J = I J.
Answer. (not given in class) a
I = n: an I, but then an J as I J, then a J.
I + J I + J.
(ii)
Ans

Answers to exercises for 6
[revised v1 28/03/2014]
1
Answers to exercises for 6
E6.1. Let R be a domain and r R be irreducible.
(i) Prove: if s r, then s is also irreducible.
(ii) View r as a polynomial of degree 0 in R[X]. Explain why r is irreducible in

Answers to exercises for 2
[revised v2 21/02/2014]
1
Answers to exercises for 2
E2.1. For arbitrary p N, construct a set S M(X, Y) such that S has exactly p minimal
elements with respect to |.
Answer. Consider for example S = cfw_Xp1 , Xp2 Y, Xp3 Y 2 , .

Answers to exercises for 4
[revised v1 27/02/2014]
1
Answers to exercises for 4
E4.1. Let F = cfw_XY XZ, X + YZ, YZ Z2 , X + Z2 . In each of the following situations, reduce
the polynomial X2 with respect to F:
(i) in Q[X, Y, Z] with respect to Lex with X

Answers to exercises for 3
[revised v2 03/03/2014]
1
Answers to exercises for 3
E3.1. Prove: every eld K is a principal ideal ring.
Answer (not given in class): Observe that K has exactly two ideals, cfw_0 and K. This is
because if I = cfw_0 is an ideal o