Math 654
Homework #5
March 7, 2013
Due Thursday, March 21, in class.
Justify all of your work.
E1. Let R be a commutative ring with 1 6= 0. Let M and N be left R-modules such that M is
finitely generated and N is noetherian. Show that M ⊗R N is noetherian.
E2. Let V be a finite dimensional vector space over a field K, and let A : V → V be a linear
transformation. For f ∈ K[x] and v ∈ V , define
f · v := (f (A))(v).
That is, substitute A into f and evaluate the resulting transformation on v. For example, if
f = x2 + 4, then
f · v = (A2 + 3 · Id)(v) = A(A(v)) + 4v.
(a) Verify that V is a K[x]-module via the above definition.
(b) Show that V is a finitely generated torsion K[x]-module.
(c) As an example, suppose that K = R and V = R2 , and A : V → V is defined by
0 2
· v.
A(v) =
−2 0
(Here we consider v to be a 2 × 1 column vector with real entries.) Find a polynomial
q ∈ R[x] so that
V ∼
= R[x]/(q(x))
as R[x]-modules.
E3. This is a continuation of E2, and the notation there remains in use.
(a) Show that there is a unique monic polynomial qA ∈ K[x] of least degree for which
qA (A) = 0. Determine qA in terms of the elementary divisors of the K[x]-module V .
(b) Let pA (x) = det(x · Id −A) ∈ K[x] be the characteristic polynomial of A. Relate pA to
the elementary divisors of V .
(c) Show that pA (A) = 0. (Thus a linear transformation satisfies its own characteristic
polynomial, which is also known as the Cayley-Hamilton Theorem.)
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