Study Guide for Final Exam

Module Theory Problems

Problem 1

Suppose $R$ is a ring, $M$ is a left $R$ -module and $N_{1} \subseteq N_{2} \subseteq \dots$ is an ascending chain of submodules of $M$ . Prove that the set $⋃_{i = 1}^{\infty} N_{i}$ is a submodule of $M$ .

Problem 2

Let $R$ be a ring, $M_{1}$ and $M_{2}$ be left $R$ -modules, and $A_{1} \subseteq M_{1}$ and $A_{2} \subseteq M_{2}$ be submodules. Prove that $A_{1} \oplus A_{2}$ is (isomorphic to) a submodule of $M_{1} \oplus M_{2}$ and that there is an $R$ -module isomorphism

$(M_{1} \oplus M_{2}) / (A_{1} \oplus A_{2}) ≃ (M_{1} / A_{1}) \oplus (M_{2} / A_{2}) .$

Problem 3

Suppose $R$ is a PID and $M$ is an $R$ -module annihilated by a nonzero proper ideal $⟨ a ⟩ \subset R$ . Let $a = p_{1}^{α_{1}} \dots p_{k}^{α_{k}}$ be the prime factorization of $a$ in $R$ , and let $M_{i} = {m \in M ∣ p_{i}^{α_{i}} m = 0}$ be the annihilator of $p_{i}^{α_{i}}$ in $M$ . Prove that there is a direct sum $R$ -module decomposition

$M = M_{1} \oplus \dots \oplus M_{k} .$

The submodule $M_{i}$ is called the $p_{i}$ -primary component of $M$ .

Hints

Use induction on $k$ (the number of distinct prime factors). First note that $M_{i}$ is the annihilator of the ideal $⟨ p_{i}^{α_{i}} ⟩ \subseteq R$ , so by a previous exercise it is a submodule of $M$ . Show that $M = M_{1} \oplus N$ , where $N$ is the annihilator of the ideal $⟨ b ⟩ \subseteq R$ , where $b = \frac{a}{p_{1}^{α_{1}}}$ . (Note that $b$ and $p_{1}^{α_{1}}$ are relatively prime in $R$ , so one has $r b + s p_{1}^{α_{1}} = 1_{R}$ for some $r, s \in R$ .)

A minor note: you should verify that the annihilator in $N$ of $⟨ p_{i}^{α_{i}} ⟩$ is the same as the annihilator in $M$ of $⟨ p_{i}^{α_{i}} ⟩$ .

Problem 4

Let $A$ be a finite abelian group of order $n > 1$ , let $p$ be a prime divisor of $n$ , and let $p^{α}$ be the largest power of $p$ dividing $n$ . Prove that $Z_{p^{α}} \otimes_{Z} A$ is isomorphic to the Sylow $p$ -subgroups of $A$ .

Hints

The abelian group $A$ is equal to the direct sum of its $p_{i}$ -primary components

$A = A_{1} \oplus \dots \oplus A_{k} .$

This is true by the Fundamental Theorem for Finite Abelian Groups (Elementary Divisor Form), which is a special case of our Fundamental Theorem for Finitely Generated Modules over a PID (in the case the ring is $Z$ ). It's also true by another exercise (in the special case $R = Z$ , $M = A$ , and $a = n$ ).

In any case, you may use the fact that the $p_{i}$ -primary component $A_{i}$ is the Sylow $p_{i}$ -subgroup of $A$ .

With this in mind, show that $Z_{p^{α}} \otimes_{Z} A_{i}$ is $0$ when $p \neq p_{i}$ and is isomorphic to $A_{i}$ when $p = p_{i}$ . Then use the fact that tensor product commutes with direct sums.

Problem 5

Prove there is a ring isomorphism $R \otimes_{Z} Z [i] ≃ C$ .

Problem 6

Suppose $R$ is an integral domain and $I \subseteq R$ is a principal ideal. Prove that the $R$ -module $I \otimes_{R} I$ has no nonzero torsion elements.

Hints

One option is to prove $I \otimes_{R} I ≃ R$ as $R$ -modules. (Since $R$ is an integral domain it is torsion free as an $R$ -module, so this proves the desired result.) To establish the desired isomorphism, let $a$ be a generator for $I$ and define the set map $I \times I \to R$ by $(r a, s a) \mapsto r s$ . Verify the necessary properties to conclude that the map $I \otimes_{R} I \to R$ defined on simple tensors by $r a \otimes s a \mapsto r s$ is a well-defined $R$ -module morphism. Then show that the $R$ -module morphism $R \to I \otimes_{R} I$ defined by $r \mapsto r (a \otimes a) = (r a) \otimes a = a \otimes r a$ is the inverse morphism.

You could also try a direct approach, but be wary of the following trap: it's very difficult to decide when a tensor equals zero. In other words, if $m \otimes n$ is zero in some tensor product $M \otimes_{R} N$ , there is in general no easy deduction one can make about $m$ or $n$ . (The only general conclusion one can make is that for every balanced bilinear map $g : M \times N \to P$ one has $g (m, n) = 0$ .)

Problem 7

Find all possible rational canonical forms of $6 \times 6$ matrices over $Q$ with characteristic polynomial $c (x) = x^{2} (x^{2} + 1)^{2}$ .

Problem 8

Determine the Jordan canonical form for the $n \times n$ matrix over $Q$ whose entries are all equal to $1$ .

Problem 9

Prove there are no $3 \times 3$ matrices $A$ over $Q$ of order 8.

Problem 10

Determine all similarity classes of $2 \times 2$ matrices over $Q$ of order $4$ .

Category Theory Problems

Problem 11

Suppose $C$ is a category and $f, g : a \to b$ are parallel morphisms in $C$ for which an equalizer $Eq (f, g) \overset{e}{\to} a$ exists. Prove that the arrow $e$ is a monomorphism in $C$ .

Problem 12

Suppose $C$ is a category that has pullbacks and a terminal object, $t$ .

Prove that $C$ has all (binary) products.
Prove that $C$ has all equalizers.

Hints

Show that for every pair of objects $a, b$ in $C$ , the pullback $a \times_{t} b$ satisfies the universal property of the product $a \times b$ .
Show that the equalizer of a pair of arrows $f, g : a \to b$ may be constructed as the pullback of $a \overset{(1_{a}, f)}{\to} a \times b \overset{(1_{a}, g)}{\leftarrow} a$ .

Problem 13

Suppose $C$ is a category that has all equalizers.

The map that assigns to each pair of parallel arrows $f, g : a \to b$ the equalizer object $Eq (f, g)$ is the object function of a functor to $C$ . What is the arrow function of that functor?
The equalizer functor above is right adjoint to a certain diagonal (or constant) functor from $C$ . Describe:
- a) the other functor;
- b) the natural bijection of the adjunction; and
- c) the unit and counit of the adjunction.

Problem 14

Let $X = {x_{1}, x_{2}, \dots}$ be any infinite countable set and let $M = F (X)$ be the free $Z$ -module (i.e., free abelian group) on $X$ . Consider the following four set maps from $X$ to $M$ , where for simplicity we simply list the images of the elements of $X$ :
$\begin{aligned} ϕ_{1} : & (x_{1}, x_{2}, x_{3}, \dots) \mapsto (x_{1}, x_{3}, x_{5}, \dots) \\ ϕ_{2} : & (x_{1}, x_{2}, x_{3}, \dots) \mapsto (x_{2}, x_{4}, x_{6}, \dots) \\ ψ_{1} : & (x_{1}, x_{2}, x_{3}, \dots) \mapsto (x_{1}, 0, x_{2}, 0, \dots) \\ ψ_{2} : & (x_{1}, x_{2}, x_{3}, \dots) \mapsto (0, x_{1}, 0, x_{2}, \dots) \end{aligned}$
Let $p_{1}, p_{2}, i_{1}, i_{2} : M \to M$ be the corresponding $Z$ -module morphisms.

Prove that the diagram below is a biproduct diagram in $A b$ :
Why does this prove $M ≃ M \oplus M$ ?

Problem 15

Suppose $f : A \to B$ is a morphism of abelian groups. Prove that the projection morphism $π : B \to B / im (f)$ is a cokernel of $f$ .

Problem 16

Suppose the diagram below is part of a double-complex in an abelian category and is vertically exact at $B$ ; i.e., $\ker (f) = 0$ :

Use the Salamander Lemma to prove that $^{◻} A ≃ A^{vert}$ and $A^{hor} ≃ A_{◻}$ .