MAS439: Problem Sheet 3
1. Let R be a commutative ring, and let U and V be multiplicative subsets
of R.
(i) Show that U V is a multiplicative subset of R.
Solution We must show that 1 U V , and that U V is closed
under multiplication. Since 1 U and 1 V we
MAS439: Problem Sheet 5
1. Check Nakayamas lemma (Version 1) by hand where we take our ring R
to be Z, our module M to be Z/10Z, and our ideal I to be (7).
In other words, show that IM = M , and exhibit any element x 1 + (7)
with xM = 0.
(3 marks)
Solutio
MAS439: Problem Sheet 1
1. Let R be the ring R[t]/(t2 1).
(i) Let V be a vector space over R, and let T be a linear involution (that
is, a linear map V V such that T 2 is the identity on V ).
Show that there is an R-module structure on V for which tv = T
MAS439: Problem Sheet 4
1. Whats the Jacobson radical of the following rings:
(a) Z?
(b) the localisation Z(11) ?
(c) the ring Z/20Z?
(3 marks)
Solution
(a) The maximal ideals of Z are (p) where p is prime. There are no
elements in common between all thes
MAS439: Problem Sheet 2
1. Let R be the ring Z/4Z of integers modulo 4.
(i) Show that there is exactly one R-module, N , with two elements.
Solution Let the elements be cfw_0, a. We must have 2a = a + a = 0.
Hence we have 3a = a + a + a = a, and this desc
MAS439: Problem Sheet 7
1. The real numbers 3 and 7 are both integral over Z, being roots respectively of x2 3 and x2 7. Its a consequence of Corollary 13.2 that their
sum 3 + 7 is also integral over Z.
Prove this directly, by exhibiting a monic polynom
MAS439: Problem Sheet 6
1. Let Z3 Z3 be the map of Z-modules dened by the matrix
2 4
4 0
0 2
2
2 .
6
Find a monic polynomial equation satised by with all coecients (except the leading coecient) divisible by 2.
(3 marks)
Solution
We compute
2
det 4
0
4
2
MAS439: Problem Sheet 8
Due: Friday, 2nd May, 4pm
1. The eld R is not algebraically closed. Give a maximal ideal I of R[x]
which is not of the form (x a) for any a R. Prove that the ideal is
maximal by identifying the quotient eld R[x]/I as a well-known e
MAS439: Problem Sheet 7
Due: Friday, 4th April, 4pm
(Corrected slightly on 1st April)
1. The real numbers 3 and 7 are both integral over Z, being roots respectively of x2 3 and x2 7. Its a consequence of Corollary 13.2 that their
sum 3 + 7 is also integ
MAS439: Problem Sheet 2
Due: Friday, 28th February, 4pm
1. Let R be the ring Z/4Z of integers modulo 4.
(i) Show that there is exactly one R-module, N , with two elements.
(ii) Show that there are exactly two dierent R-modules, M1 and M2 ,
with four eleme
MAS439: Problem Sheet 1
Due: Friday, 21st February, 4pm
1. Let R be the ring R[t]/(t2 1).
(i) Let V be a vector space over R, and let T be a linear involution (that
is, a linear map V V such that T 2 is the identity on V ).
Show that there is an R-module
MAS439: Problem Sheet 4
Due: Friday, 14th March, 4pm
1. Whats the Jacobson radical of the following rings:
(a) Z?
(b) the localisation Z(11) ?
(c) the ring Z/20Z?
(3 marks)
2. Let U Z be the set of integers which are not multiples of 3 or 7; in other
word
MAS439: Problem Sheet 3
Due: Friday, 7th March, 4pm
1. Let R be a commutative ring, and let U and V be multiplicative subsets
of R.
(i) Show that U V is a multiplicative subset of R.
(ii) Show that the set cfw_uv u U, v V is a multiplicative subset of R.
MAS439: Problem Sheet 9
Due: Friday, 9th May, 4pm
1. Projectivise the following varieties, and nd all the points at innity:
(a) V (J1 ), where J1 = (xy) is an ideal in C[x, y];
(b) V (J2 ), where J2 = (x2 + y 2 1) is an ideal in R[x, y];
(c) V (J3 ), wher
MAS439: Problem Sheet 5
Due: Friday, 21st March, 4pm
1. Check Nakayamas lemma (Version 1) by hand where we take our ring R
to be Z, our module M to be Z/10Z, and our ideal I to be (7).
In other words, show that IM = M , and exhibit any element x 1 + (7)
w
MAS439: Problem Sheet 6
Due: (Unusually!) Tuesday, April 1st, 4pm
1. Let Z3 Z3 be the map of Z-modules dened by the matrix
2 4
4 0
0 2
2
2 .
6
Find a monic polynomial equation satised by with all coecients (except the leading coecient) divisible by 2.
(3