Pries: 567 Abstract Algebra
Midterm: Monday, March 5, 2012
Name:
Read the whole exam carefully before you begin. Show all your work and explain all your
answers to receive full credit.
1. Let R = Z. What is the annihilator of the R-module M = Z/15 ⊗Z Z/10 in R?
2. What is the bilinear (R-balanced) map that is useful for proving that S ⊗R Rn ' S n ?
3. Let V be the vector space of polynomials f (x) ∈ Q[x] with deg(f (x)) ≤ 3 and basis
v1 = 1, v2 = x, v3 = x2 , v4 = x3 . The map φ : V → Q given by φ(f (x)) = f 0 (5) is an
element of the dual space of V . Express it as a linear combination of elements of the
dual basis v1∗ , v2∗ , v3∗ , v4∗ .
4. Let R = k[x]. Let M = k[x] be the R-module where the action is multiplication. Is
the map φ : M → M given by φ(f (x)) = x2 f (x) an R-module homomorphism?
5. Let V be a vector space over a field K. Let T : V → V be a linear transformation such
that T 2 = T (a projection map). If V is a cyclic K[x]-module, prove that dim(V ) ≤ 2.
6. Let {e1 , e2 } be a basis of V = R2 . Show that the element e1 ⊗ e2 + e2 ⊗ e1 in V ⊗R V
cannot be written as a simple tensor v ⊗ w for any v, w ∈ R2 .
7. Let V = {f : R → R | f continuous}. We say that f is even if f (−x) = f (x) for all
x ∈ R. We say that f is odd if f (−x) = −f (x) for all x ∈ R. Let W ⊂ V be the
subspace consisting of all even functions f . Let W 0 ⊂ V be the subspace consisting of
all odd functions f . Let π : V → V be the linear transformation
π(f ) =
f (x) + f (−x)
.
2
You may use the facts that π(f ) ∈ W and f − π(f ) ∈ W 0 .
(a) Show that W ∩ W 0 = {0}.
(b) Show that Span(W, W 0 ) = V .
(c) Why can you conclude that V ' W ⊕ W 0 ?
(d) Show that Im(π) = W .
(e) Show that Ker(π) = W 0 .
(f) Explain why this is a split exact sequence:
inc
π
0 → W 0 → V → W → 0.
Take-home problem: Feel free to refer to books, notes, or webpages, but do not discuss
the problem with other people.
Hand in under my office door by Thursday 3/15.
Let D, L, M , and N be R-modules. Suppose there is an exact sequence of R-modules:
ψ
φ
0 → L → M → N → 0.
In this problem, you do not need to check whether maps are R-module homomorphisms.
1. Show that there is an injective homomorphism:
φ0 : HomR (N, D) → HomR (M, D).
2. Show that there is a homomorphism:
ψ 0 : HomR (M, D) → HomR (L, D),
such that
Im(φ0 ) = Ker(ψ 0 ).
3. Conclude that there is an exact sequence:
φ0
ψ0
0 → HomR (N, D) → HomR (M, D) → HomR (L, D).
4. Give an example to show that ψ 0 does not need to be surjective.