Homework 2 Due: 5th March, 2012 Flat modules 1. Let R be a ...

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