Math 654
Homework #4
February 21, 2013
Due Thursday, March 7, in class.
Justify all of your work.
Terminology: Functions that Hungerford calls “middle linear maps” in section IV.5 are what we
have been calling R-biadditive maps in class.
Note: One of the problems that I mentioned in class turned out to be Theorem 5.8 in the text, so
it is no long assigned.
D1. Show that Z/mZ ⊗Z Z/nZ ∼
= Z/tZ for some t ∈ Z. What is t?
D2. Show that Q ⊗Z Q ∼
6 R as an R-vector space.
= Q as a Q-vector space, but that R ⊗Q R ∼
=
D3. Let R be a commutative ring (with 1), and let M , N , and P be R-modules. Show that
HomR (M ⊗R N, P ) ∼
= HomR (M, HomR (N, P ))
as R-modules.
D4. Let K be a commutative ring (with 1), and suppose that A and B are K-algebras. We can
form A ⊗K B, which is a K-module, but it also has the structure of a K-algebra via
(a1 ⊗ b1 ) · (a2 ⊗ b2 ) := (a1 a2 ) ⊗ (b1 b2 ),
extended linearly to a binary operation on A ⊗K B. If A and B are both commutative rings,
then so is A⊗K B. (For more details, be sure to read about this in Theorem 7.4 in section IV.7
of Hungerford. You can of course take the content of section IV.7 as given for this problem.)
Now for d ∈ Z, not a perfect square, consider the field
√
√
Q( d) = {a + b d | a, b ∈ Q}.
Let p and q be primes.
(a) Define a surjective Q-algebra homomorphism
√
√
√
φ : Q( p)[X] → Q( p) ⊗Q Q( q).
√
√
(b) If p 6= q, use part (a) to show that Q( p) ⊗Q Q( q) is a field.
√
√
(c) Show that Q( p) ⊗Q Q( p) is not an integral domain by finding a zero divisor. What
is the kernel of φ in this case?
1