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 $A_{1} ⨁ A_{2}$ is (isomorphic to) a submodule of $M_{1} ⨁ M_{2}$ and that there is an $R$ -module isomorphism

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

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

$M = M_{1} ⨁ \dots ⨁ M_{k} .$

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 an abelian group of order $n = p_{1}^{m_{1}} \dots p_{k}^{m_{k}} > 1$ . It is a fact that for each prime divisor $p_{i}$ of $n$ the Sylow $p_{i}$ -subgroup of $A$ is $A_{p_{i}} = {a \in A ∣ p_{i}^{m_{i}} \cdot a = 0_{A}}$ .

Prove that $A ≃ A_{p_{1}} ⨁ \dots ⨁ A_{p_{k}}$ .
Prove that for each prime $p_{i}$ we have $Z_{p_{i}^{m_{i}}} ⨂_{Z} A ≃ A_{p_{i}}$ .

Problem 5

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

Problem 6

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

Hints

One option is to prove $I ⨂_{R} I ≃ R$ 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 $I ⨂_{R} I \to R$ 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 $R \to I ⨂_{R} I$ 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 $M ⨂_{R} N$ , 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 arrows 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 and $f, g : a \to b$ are parallel arrows in $C$ for which a coequalizer $b \overset{h}{\to} Coeq (f, g)$ exists. Prove that the arrow $h$ is an epimorphism in $C$ .

Problem 13

Let $C$ be a category, let $i_{C} : 0 \to C$ denote the unique functor from the initial category $0$ to $C$ , and let $z : 0 \to 1$ denote the unique functor from the initial category $0$ to the terminal category $1$ .

Show that the right Kan extension of $i_{C}$ along $z$ exists if and only if $C$ has an terminal object, and that when either exists the functor ${Ran}_{z} (i_{C}) : 1 \to C$ is the functor corresponding to the terminal object under the equivalence $C^{1} ≃ C$ .

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 $i_{1}, i_{2}, p_{1}, p_{2} : M \to M$ be the corresponding $Z$ -module morphisms.

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

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 $A^{hor} ≃ A_{◻}$ .