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

Under construction!

This problem is currently under construction, but will be available soon.

Problem 13

Under construction!

This problem is currently under construction, but will be available soon.

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 $A^{hor} ≃ A_{◻}$ .