# Math 654 Homework #4 February 21, 2013 Due Thursday, March 7, in class. ```Math 654
Homework #4
February 21, 2013
Due Thursday, March 7, in class.
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 ) &middot; (a2 ⊗ b2 ) := (a1 a2 ) ⊗ (b1 b2 ),
extended linearly to a binary operation on A ⊗K B. If A and B are both commutative rings,
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
```