Homework 2 Due: 5th March, 2012 Flat modules 1. Let R be a ...
Homework 2 Due: 5th March, 2012 Flat modules 1. Let R be a ...
Homework 2 Due: 5th March, 2012 Flat modules 1. Let R be a ...
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<strong>Homework</strong> 2<br />
<strong>Due</strong>: 5 th <strong>March</strong>, <strong>2012</strong><br />
<strong>Flat</strong> <strong>modules</strong><br />
<strong>1.</strong> <strong>Let</strong> R <strong>be</strong> a commutative ring.<br />
a) If M and N are flat R-<strong>modules</strong>, then so is M ⊗ R N.<br />
b) If S is a flat R-algebra and M is a flat S-module, then M is flat as an R-module.<br />
2. <strong>Let</strong> R <strong>be</strong> commutative. An R-module M is said to <strong>be</strong> faithfully flat if it is flat and if M ⊗ R N = 0,<br />
then N = 0. A homomorphism of commutative rings φ : R → S is said to <strong>be</strong> (faithfully) flat when S<br />
is (faithfully) flat as an R-module. Show the following:<br />
a) <strong>Let</strong> φ : R → S <strong>be</strong> faithfully flat. An R-module M is (faithfully) flat if S ⊗ A M is (faithfully) flat as<br />
an S-module.<br />
b) <strong>Let</strong> ψ : R → R ′ <strong>be</strong> a homomorphism of commutative rings. If φ : R → S is (faithfully) flat, then so<br />
is the induced map R ′ → S ⊗ R R ′ .<br />
Hil<strong>be</strong>rt’s Third Problem<br />
3. Show that if cos 2πm<br />
n<br />
∈ Q, then it is equal to one of the values: 0, ± 1 2<br />
, ±<strong>1.</strong> In particular, show that<br />
cos −1 1 3<br />
/∈ Qπ.<br />
(Hint: Suppose cos 2πm<br />
n<br />
∈ Q and (m, n) = <strong>1.</strong> <strong>Let</strong> ζ n = cos 2πm 2πm<br />
n<br />
+ i sin<br />
n<br />
. Given your assumption<br />
that cos 2πm<br />
n<br />
∈ Q, what can you say about the degree of the field extension [Q(ζ) : Q]? On the other<br />
hand, note that ζ is a primitive n th root of unity. What can you say about the degree of the extension<br />
[Q(ζ) : Q] in terms of the prime factors of n? Use the fact that these two quantities must <strong>be</strong> equal to<br />
show that the only possible values of n are 1, 2, 3, 4, 6.)<br />
Division Rings<br />
4. a) If D is a division ring, show that a finite subgroup of the units D × need not <strong>be</strong> cyclic.<br />
b) If D is a division ring whose centre is a field of characteristic p > 0, show that every finite subgroup<br />
G of the units D × is cyclic.<br />
(Hint: b) Use Wedderburn’s Theorem.)<br />
Chain Conditions and the Jacobson radical<br />
5. a) Show that the following is an example of a ring that is right Artinian but not left Artinian. The<br />
underlying set of the ring R is ( ) a b<br />
{ |a ∈ Q; b, c ∈ R}<br />
0 c<br />
and the operations are the usual operations of matrix addition and matrix multiplication.<br />
b) Show that the following is an example of a ring that is left Noetherian but not right Noetherian.<br />
The underlying set of the ring S is<br />
( a 0<br />
{<br />
b c<br />
)<br />
|a ∈ Z; b, c ∈ Q}<br />
c) <strong>Let</strong> X <strong>be</strong> a compact infinite Hausdorff topological space. <strong>Let</strong> C 0 (X, R) denote the ring of continuous<br />
real-valued functions on X. Show that this ring is not Noetherian.<br />
(Hint: a) Use the fact that there are only finitely many right ideals in R. On the other hand a subset<br />
of R of the form<br />
( ) 0 v<br />
{ |v ∈ V }<br />
0 0<br />
where Q ⊂ V ⊂ R is a vector space, is a left ideal. )<br />
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6. <strong>Let</strong> R <strong>be</strong> a commutative ring where the zero ideal can <strong>be</strong> expressed as a product of (not necessarily<br />
distinct) maximal ideals m 1 m 2 · · · m k . Show that R is Noetherian if and only if R is Artinian.<br />
7. <strong>Let</strong> R <strong>be</strong> a commutative ring and let M n (R) denote the ring of n × n matrices with coeffecients in R.<br />
Show that the following are equivalent:<br />
(a) R is Noetherian.<br />
(b) For some n, the ring M n (R) has ascending chain condition on (2-sided) ideals.<br />
(c) For all n, the ring M n (R) has ascending chain condition on (2-sided) ideals.<br />
8. <strong>Let</strong> R <strong>be</strong> a commutative Noetherian local ring with unique maximal ideal m. Show that<br />
∩ n≥1 m n = 0.<br />
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