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An Introduction to Homological Algebra

Joseph J. Rotman

Chapter 3

Special Modules - all with Video Answers

Educators

Chapter Questions

00:45

Problem 1

Let $M$ be a free $R$-module, where $R$ is a domain. Prove that if $r m=0$, where $r \in R$ and $m \in M$, then either $r=0$ or $m=0 .$ (This is false if $R$ is not a domain.)

Amy Jiang
Amy Jiang
Numerade Educator
08:10

Problem 2

Let $R$ be a ring and let $S$ be a nonzero submodule of a free right $R$-module $F$. Prove that if $a \in R$ is not a right zero-divisor $^{2}$, then $S a \neq\{0\}$.

Vishvajeetkumar Bhaskar Batule
Vishvajeetkumar Bhaskar Batule
Numerade Educator
08:50

Problem 3

Define projectivity in Groups, and prove that a group $G$ is projective if and only if $G$ is a free group.
Hint. Recall the Nielsen-Schreier Theorem: Every subgroup of a free group is free.

Ely Crowder
Ely Crowder
Numerade Educator
01:57

Problem 4

4 (i) (Pontrjagin) If $A$ is a countable torsion-free abelian group each of whose subgroups $S$ of finite rank is free abelian, prove that $A$ is free abelian (the $\operatorname{rank}$ of an abelian group $S$ is defined as $\operatorname{dim}_{\mathbb{Q}}\left(\mathbb{Q} \otimes_{\mathbb{Z}} S\right)$; cf. Exercise $2.36$ on page 97).
Hint. See the discussion on page 103 .
(ii) Prove that every subgroup of finite rank in $\mathbb{Z}^{\mathbb{N}}$ (the product of countably many copies of $\mathbb{Z}$ ) is free abelian.
(iii) Prove that every countable subgroup of $\mathbb{Z}^{\mathbb{N}}$ is free. (In Theorem $4.17$, we will see that $\mathbb{Z}^{\mathbb{N}}$ itself is not free.)

Wendi Zhao
Wendi Zhao
Numerade Educator
01:21

Problem 5

(Eilenberg) Prove that every projective left $R$-module $P$ has a free complement; that is, there exists a free left $R$-module $F$ such that $P \oplus F$ is free.
Hint. If $P \oplus Q$ is free, consider $Q \oplus P \oplus Q \oplus P \oplus \cdots$.

Jay Patel
Jay Patel
Numerade Educator
03:13

Problem 6

Let $k$ be a commutative ring, and let $P$ and $Q$ be projective $k$ modules. Prove that $P \otimes_{k} Q$ is a projective $k$-module.

Gideon Idumah
Gideon Idumah
Numerade Educator
01:16

Problem 7

(i) Prove that $R=C(\mathbb{R})$, the ring of all real-valued functions on $\mathbb{R}$ under pointwise operations, is not noetherian.
(ii) Recall that $f: \mathbb{R} \rightarrow \mathbb{R}$ is a $C^{\infty}$-function if $\partial^{n} f / \partial x^{n}$ exists and is continuous for all $n$. Prove that $R=C^{\infty}(\mathbb{R})$, the ring of all $C^{\infty}$-functions on $\mathbb{R}$ under pointwise operations, is not noetherian.
(iii) If $k$ is a commutative ring, prove that $k[X]$, the polynomial ring in infinitely many indeterminates $X$, is not noetherian.

Carson Merrill
Carson Merrill
Numerade Educator
02:33

Problem 8

(Small) Let $R$ be the ring of all $2 \times 2$ matrices $\left[\begin{array}{ll}a & 0 \\ b & c\end{array}\right]$ with $a \in \mathbb{Z}$ and $b, c \in \mathbb{Q}$ is a ring. Schematically, we can describe $R$ as $\left[\begin{array}{ll}\mathbb{Z} & 0 \\ \mathbb{Q} & \mathbb{Q}\end{array}\right]$. Prove that $R$ is left noetherian, but that $R$ is not right noetherian.

Shahab Ullah
Shahab Ullah
Numerade Educator
03:28

Problem 9

Let $V$ be a vector space over a field $k$.
(i) Prove that $V$ is a free $k$-module.
(ii) Prove that a subset $B$ of $V$ is a basis of $V$ considered as a vector space if and only if $B$ is a basis of $V$ considered as a free $k$-module.

Mengchun Cai
Mengchun Cai
Numerade Educator
05:58

Problem 10

(i) If $R$ is a domain and $I$ and $J$ are nonzero ideals in $R$, prove that $I \cap J \neq\{0\}$.
(ii) Let $R$ be a domain and let $I$ be an ideal in $R$ that is a free $R$-module; prove that $I$ is a principal ideal.

Vishvajeetkumar Bhaskar Batule
Vishvajeetkumar Bhaskar Batule
Numerade Educator
06:40

Problem 11

Prove that $\operatorname{Hom}_{R}(P, R) \neq\{0\}$ if $P$ is a nonzero projective left $R$ module.

John Gehad
John Gehad
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01:31

Problem 12

If $P$ is a finitely generated left $R$-module, prove that $P$ is projective if and only if $1_{P} \in \operatorname{im} v$, where $v: \operatorname{Hom}_{R}(P, R) \otimes_{R} P \rightarrow$ $\underset{\sim}{\operatorname{Hom}}_{R}(P, P)$ is defined, for all $x \in P$, by $f \otimes x \mapsto \widetilde{f}$, where $\widetilde{f}: y \mapsto f(y) x .$
Hint. Use a projective basis.

Carson Merrill
Carson Merrill
Numerade Educator
07:15

Problem 13

Let $R$ be a commutative ring, and let $A$ and $B$ be finitely generated $R$-modules.
(i) Prove that $A \otimes_{R} B$ is a finitely generated $R$-module.
(ii) If $R$ is noetherian, prove that $\operatorname{Hom}_{R}(A, B)$ is a finitely generated $R$-module.
(iii) Give an example showing that $\operatorname{Hom}_{R}(A, B)$ may not be finitely generated if $R$ is not noetherian.
Hint (Griffith). Let $V$ be an infinite-dimensional vector space over $\mathbb{F}_{p}$, and let $R=\mathbb{Z} \oplus V$, where $(m, v)\left(m^{\prime}, v^{\prime}\right)=$ $\left(m m^{\prime}, m v^{\prime}+m^{\prime} v\right)$. Then $V$ is an ideal in $R$ that is not finitely generated, and if $A=(R / V) / p(R / V)$, then $\operatorname{Hom}_{R}(A, R) \cong V$ as $R$-modules.

Chris Trentman
Chris Trentman
Numerade Educator
01:05

Problem 14

Prove the dual of Schanuel's Lemma. Given exact sequences $0 \rightarrow M \stackrel{i}{\rightarrow} E \stackrel{p}{\rightarrow} Q \rightarrow 0 \quad$ and $\quad 0 \rightarrow M \stackrel{i^{\prime}}{\rightarrow} E^{\prime} \stackrel{p^{\prime}}{\rightarrow} Q^{\prime} \rightarrow 0$ where $E$ and $E^{\prime}$ are injective, then there is an isomorphism
$$
Q \oplus E^{\prime} \cong Q^{\prime} \oplus E .
$$

Anthony Ramos
Anthony Ramos
Numerade Educator
01:21

Problem 15

(Schanuel) Let $B$ be a left $R$-module over some ring $R$, and consider two exact sequences
$$
0 \rightarrow K \rightarrow P_{n} \rightarrow P_{n-1} \rightarrow \cdots \rightarrow P_{1} \rightarrow P_{0} \rightarrow B \rightarrow 0
$$ $$
0 \rightarrow K^{\prime} \rightarrow Q_{n} \rightarrow \mathbb{Q}_{n-1} \rightarrow \cdots \rightarrow Q_{1} \rightarrow Q_{0} \rightarrow B \rightarrow 0
$$
where the $P$ s and $Q$ s are projective. Prove that
$$
K \oplus Q_{n} \oplus P_{n-1} \oplus \cdots \cong K^{\prime} \oplus P_{n} \oplus Q_{n-1} \oplus \cdots .
$$

Jay Patel
Jay Patel
Numerade Educator
13:29

Problem 16

Let $R$ be a ring with IBN.
(i) If $0 \rightarrow F_{n} \rightarrow \cdots \rightarrow F_{0} \rightarrow 0$ is an exact sequence with each $F_{i}$ a finitely generated free $R$-module, prove that $\sum_{i=0}^{n}(-1)^{i} \operatorname{rank}\left(F_{i}\right)=0$.
(ii) Let $0 \rightarrow F_{n} \rightarrow \cdots \rightarrow F_{0} \rightarrow M \rightarrow 0$ and $0 \rightarrow F_{m}^{\prime} \rightarrow$ $\cdots \rightarrow F_{0}^{\prime} \rightarrow M \rightarrow 0$ be exact sequences of left $R$ modules, where each $F_{i}$ and $F_{j}^{\prime}$ is finitely generated and free. Prove that
$$
\sum_{i=0}^{n}(-1)^{i} \operatorname{rank}\left(F_{i}\right)=\sum_{j=0}^{m}(-1)^{j} \operatorname{rank}\left(F_{j}^{\prime}\right) .
$$
The common integer value is called the Euler characteristic of $M$ and is denoted by $\chi(M)$.
Hint. Use Exercise 3.15.
(iii) Let $0 \rightarrow M^{\prime} \rightarrow M \rightarrow M^{\prime \prime} \rightarrow 0$ be an exact sequence of finitely generated left $R$-modules. If two of the modules have an Euler characteristic, prove that the third does also, and
$$
\chi(M)=\chi\left(M^{\prime}\right)+\chi\left(M^{\prime \prime}\right) .
$$
Hint. Use Corollary $3.13 .$

Anthony Ramos
Anthony Ramos
Numerade Educator
14:08

Problem 17


(i) Prove that every vector space over a field $k$ is an injective $k$-module.
(ii) Prove that if $0 \rightarrow U \rightarrow V \rightarrow W \rightarrow 0$ is an exact sequence of vector spaces, then the corresponding sequence of dual spaces $0 \rightarrow W^{*} \rightarrow V^{*} \rightarrow U^{*} \rightarrow 0$ is also exact.

Anthony Ramos
Anthony Ramos
Numerade Educator
03:58

Problem 18


(i) Prove that if a domain $R$ is an injective $R$-module, then $R$ is a field.
(ii) Prove that $\mathbb{I}_{6}$ is simultaneously an injective and a projective module over itself.
(iii) Let $R$ be a domain that is not a field, and let $M$ be an $R$-module that is both injective and projective. Prove that $M=\{0\}$.
Hint. Use Exercises $2.22$ on page 68 and $3.11$ on page 115 .

Anthony Ramos
Anthony Ramos
Numerade Educator
01:02

Problem 19

(Pontrjagin Duality) If $G$ is a (discrete) abelian group, its Pontrjagin dual is the group
$$
G^{*}=\operatorname{Hom}_{\mathrm{Z}}(G, \mathbb{R} / \mathbb{Z}) .
$$
(More generally, the Pontrjagin dual of a locally compact abelian topological group $G$ consists of all the continuous homomorphisms from $G$ into the circle group $S^{1} \cong \mathbb{R} / \mathbb{Z}$.)
(i) If $G$ is an abelian group and $a \in G$ is nonzero, prove that there is a homomorphism $f: G \rightarrow \mathbb{R} / \mathbb{Z}$ with $f(a) \neq 0$.
(ii) Prove that $\mathbb{R} / \mathbb{Z}$ is an injective abelian group.
(iii) Prove that if $0 \rightarrow A \rightarrow G \rightarrow B \rightarrow 0$ is an exact sequence of abelian groups, then so is $0 \rightarrow B^{*} \rightarrow G^{*} \rightarrow A^{*} \rightarrow 0$.
(iv) If $G$ is a finite abelian group, then $G^{*} \cong \operatorname{Hom}_{\mathrm{Z}}(G, Q / \mathbb{Z})$.
(v) If $G$ is a finite abelian group, prove that $G^{*} \cong G$.
(vi) Prove that every quotient group $G / H$ of a finite abelian group $G$ is isomorphic to a subgroup of $G$.
Remark. The analogous statement for nonabelian groups is false: if $Q$ is the group of quaternions, then $Q / Z(Q) \cong$ $\mathbf{V}$, where $Z(\mathbf{Q})$ is the center of $\mathbf{Q}$ and $\mathbf{V}$ is the four-group. But $\mathbf{Q}$ has only one element of order 2 while $\mathbf{V}$ has three elements of order 2, so that $\mathbf{V}$ is not isomorphic to a subgroup of $\mathbf{Q}$. Part (vi) is also false for infinite abelian groups: since $\mathbb{Z}$ has no element of order 2 , it has no subgroup isomorphic to $\mathbb{Z} / 2 \mathbb{Z}=\mathrm{I}_{2}$.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:05

Problem 20

Being an essential extension is transitive. Let $M \subseteq E \subseteq E_{1}$ be submodules of a left $R$-module $E_{1}$. If $E$ is an essential extension of $M$ and $E_{1}$ is an essential extension of $E$, prove that $E_{1}$ is an essential extension of $M$.

Anthony Ramos
Anthony Ramos
Numerade Educator
01:53

Problem 21

(i) Let $M \subseteq E$ be left $R$-modules. Prove that $E$ is an essential extension of $M$ if and only if, for every nonzero $e \in E$, there is $r \in R$ with $r e \in M$ and $r e \neq 0$.
(ii) Let $M \subseteq E$ be left $R$-modules, and let $\mathcal{S}$ be a chain of intermediate submodules; that is, $M \subseteq S \subseteq E$ for all $S \in \mathcal{S}$ and, if $S, S^{\prime} \in \mathcal{S}$, either $S \subseteq S^{\prime}$ or $S^{\prime} \subseteq S$. If each $S \in \mathcal{S}$ is an essential extension of $M$, use part (i) to prove that $\bigcup_{S \in \mathcal{S}} S$ is an essential extension of $M$.

Anthony Ramos
Anthony Ramos
Numerade Educator
03:33

Problem 22

Let $M \subseteq E^{\prime} \subseteq E$ be left $R$-modules. If both $E^{\prime}$ and $E$ are essential extensions of $M$, prove that $E$ is an essential extension of $E^{\prime}$.

Suman Saurav Thakur
Suman Saurav Thakur
Numerade Educator
01:53

Problem 23

Let $E$ be an essential extension of a left $R$-module $M$. If $\varphi: E \rightarrow N$ is an $R$-map with $\varphi \mid M$ injective, prove that $\varphi$ is injective.
Hint. Consider $M \cap \operatorname{ker} \varphi$.

Anthony Ramos
Anthony Ramos
Numerade Educator
01:40

Problem 24

If $R$ is a domain, prove that $\operatorname{Frac}(R)=\operatorname{Env}(R)$, its injective envelope.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
06:47

Problem 25

Recall that every abelian group $G$ having no elements of infinite order has a primary decomposition: $G=\bigoplus_{p} G_{p}$, where $p$ is a prime and $G_{p}=\{g \in G:$ order $g$ is some power of $p\}$. In particular, the $p$-primary component of $G=\mathbb{Q} / \mathbb{Z}$ is called the Prüfer group; it is denoted by $\mathbb{Z}\left(p^{\infty}\right)$.
(i) Prove that $\mathbb{Z}\left(p^{\infty}\right)$ is an injective abelian group.
(ii) Prove that the injective envelope $\operatorname{Env}\left(\mathbb{I}_{p^{n}}\right)$ is $\mathbb{Z}\left(p^{\infty}\right)$.

Brandon Collins
Brandon Collins
Numerade Educator
03:38

Problem 26

(i) If $A$ is the abelian group with the presentation
$$
A=\left(a_{n}, n \geq 0 \mid p a_{0}=0, p a_{n+1}=a_{n}\right)
$$
prove that $A \cong \mathbb{Z}\left(p^{\infty}\right)$.
(ii) Give an example of two injective submodules of a module whose intersection is not injective.
Hint. Define $E=A \oplus\{0\}$ and $E^{\prime}=\left\langle\left\{\left(a_{n+1}, a_{n}\right): n \geq 0\right\}\right\rangle$ in $A \oplus A$.

Wendi Zhao
Wendi Zhao
Numerade Educator
03:59

Problem 27

Prove that $\mathbb{I}_{2}$ is not a flat $\mathbb{Z}$-module.

Aidan Mcnabb
Aidan Mcnabb
Numerade Educator
02:52

Problem 28

Let $k$ be a commutative ring, and let $P$ and $Q$ be flat $k$-modules. Prove that $P \otimes_{k} Q$ is a flat $k$-module.

Wendi Zhao
Wendi Zhao
Numerade Educator
00:40

Problem 29

Let $R$ be a PID, let $Q=\operatorname{Frac}(R)$, and let $M$ be a torsion-free $R$ module.
(i) Prove that $M$ can be imbedded in $Q \otimes_{R} M$.
(ii) Prove that $Q \otimes_{R} M \cong \operatorname{Env}(M)$, the injective envelope of $M$.

Sriparna Bhattacharjee
Sriparna Bhattacharjee
Numerade Educator
02:17

Problem 30

If $R$ is a commutative ring (not necessarily a domain), define $t M=\{m \in M: r m=0$ for some nonzero $r \in R\} .$
(i) Let $R=\mathbb{I}_{6}$, and regard $R$ as a module over itself. Prove that $[1] \notin t \mathbb{I}_{6}$.
(ii) Prove that $t I_{6}$ is not a submodule of $I_{6}$.
Hint. Both $[2],[3] \in t \mathbb{I}_{6}$, but $[3]-[2] \notin t \mathbb{I}_{6}$.

Victor Salazar
Victor Salazar
Numerade Educator
06:02

Problem 31

(i) Let $P$ be the set of all primes in $\mathbb{Z}$. Prove that $\bigoplus_{p \in P} \mathbb{I}_{p}$ is the torsion subgroup of $\prod_{p \in P} \mathbb{I}_{p}$.
(ii) Prove that $\left(\prod_{p \in P} \mathbb{I}_{p}\right) /\left(\bigoplus_{p \in P} \mathbb{I}_{p}\right)$ is divisible.
(iii) Prove that $t\left(\prod_{p \in P} \mathbb{I}_{p}\right)$ is not a direct summand of $\prod_{p \in P} \mathbb{I}_{p}$.

Ely Crowder
Ely Crowder
Numerade Educator
04:41

Problem 32

Let $0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$ be an exact sequence of right $R$ modules, for some ring $R$. If both $A$ and $C$ are flat modules, prove that $B$ is a flat module.
Hint. This result is routine if one uses the derived functor Tor.

Mohamed Mohamed
Mohamed Mohamed
Numerade Educator
02:41

Problem 33

Let $R$ be a domain, let $T$ be a torsion $R$-module, and let $D$ be a divisible $R$-module. Prove that $T \otimes_{R} D=\{0\}$. (See Proposition 2.73.)

Dharmendra Jain
Dharmendra Jain
Numerade Educator
01:05

Problem 34

Let $B=R B$, so that $\operatorname{Hom}_{R}(B, R)$ is a right $R$-module. If $C$ is a left $R$-module, define $v: \operatorname{Hom}_{R}(B, R) \otimes_{R} C \rightarrow \operatorname{Hom}_{R}(B, C)$ by $v: f \otimes c \mapsto \widehat{f}$, where $\widehat{f}(b)=f(b) c$ for all $b \in B$ and $c \in C$.
(i) Prove that $v$ is natural in $B$.
(ii) Prove that $v$ is an isomorphism if $B$ is finitely generated free.
(iii) If $B$ is a finitely presented left $R$-module and $C$ is a flat left $R$-module, prove that $v$ is an isomorphism.

Anthony Ramos
Anthony Ramos
Numerade Educator
04:47

Problem 35

A right $R$-module $B$ is called faithfully flat if
(i) $B$ is a flat module,
(ii) for all left $R$-modules $X$, if $B \otimes_{R} X=\{0\}$, then $X=\{0\}$.
Prove that $R[x]$ is a faithfully flat $R$-module (if $R$ is not commutative, then $R[x]$ is the polynomial ring in which the indeterminate $x$ commutes with each coefficient in $R$ ).

Chris Trentman
Chris Trentman
Numerade Educator
05:10

Problem 36

Prove that a right $R$-module $B$ is faithfully flat if and only if $B$ is flat and $B \otimes_{R}(R / I) \neq\{0\}$ for all proper left ideals $I$ of $R$.

Doruk Isik
Doruk Isik
Numerade Educator
05:10

Problem 37

(i) Prove that a right $R$-module $B$ is flat if and only if exactness of any sequence of left $R$-modules $A^{\prime} \stackrel{i}{\longrightarrow} A \stackrel{P}{\longrightarrow} A^{\prime \prime}$ implies exactness of $B \otimes R A^{\prime} \stackrel{1 \otimes i}{\longrightarrow} B \otimes R A \stackrel{1 \otimes p}{\longrightarrow} B \otimes R A^{\prime \prime}$.
(ii) Prove that a right $R$-module $B$ is faithfully flat if and only if it is flat and $B \otimes_{R} A^{\prime} \stackrel{1 \otimes i}{\longrightarrow} B \otimes_{R} A \stackrel{1 \otimes p}{\longrightarrow} B \otimes_{R} A^{\prime \prime}$ exact implies $A^{\prime} \stackrel{i}{\longrightarrow} A \stackrel{p}{\longrightarrow} A^{\prime \prime}$ is exact.

Doruk Isik
Doruk Isik
Numerade Educator
05:10

Problem 38

Prove that if $B$ is a faithfully flat module and $C$ is a flat module, then $B \oplus C$ is faithfully flat.

Doruk Isik
Doruk Isik
Numerade Educator
View

Problem 39

(i) Prove that $\mathbb{Q}$ is a flat $\mathbb{Z}$-module that is not faithfully flat.
(ii) Prove that an abelian group $G$ is a faithfully flat $\mathbb{Z}$-module if and only if it is torsion-free and $p G \neq G$ for all primes $p$.

Nick Johnson
Nick Johnson
Numerade Educator
03:02

Problem 40

Let $0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$ be an exact sequence of right $R$ modules, for some ring $R$. If both $A$ and $C$ are flat modules and if one of them if faithfully flat, prove that $B$ is a faithfully flat module.

Andrija Isakov
Andrija Isakov
Numerade Educator
01:05

Problem 41

Prove that if $B=R B_{S}$ is a bimodule that is $R$-flat, and if $C=C_{S}$ is $S$-injective, then $\operatorname{Hom}_{S}(B, C)$ is an injective left $R$-module.
Hint. The composite of exact functors is an exact functor.

Anthony Ramos
Anthony Ramos
Numerade Educator
04:41

Problem 42

Prove that an exact sequence $0 \rightarrow B^{\prime} \rightarrow B \rightarrow B^{\prime \prime} \rightarrow 0$ of left $R$ modules is pure exact if and only if it remains exact after tensoring by all finitely presented right $R$-modules $A$. Hint. That an element lies in $\operatorname{ker}\left(A \otimes_{R} B^{\prime} \rightarrow A \otimes_{R} B\right)$ involves only finitely many elements of $A$.

Mohamed Mohamed
Mohamed Mohamed
Numerade Educator
01:40

Problem 43

(Kulikov) If $H$ and $K$ are torsion abelian groups, prove that $H \otimes_{\mathrm{Z}} K$ is a direct sum of cyclic groups.
Hint. Use Kulikov's Theorem: if $G$ is a $p$-primary abelian group, then there exists a pure exact sequence $0 \rightarrow B \rightarrow G \rightarrow D \rightarrow$ 0 with $B$ a direct sum of cyclic groups and $D$ divisible. Such a pure subgroup $B$ is called a basic subgroup of $G$. See Rotman, An Introduction to the Theory of Groups, p. 327 .

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:02

Problem 44

If $G$ is a finite abelian group, prove that a subgroup $S \subseteq G$ is a direct summand of $G$ if and only if $S$ is a pure subgroup of $G$.
Hint. Proposition $3.71$.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:02

Problem 45

Let $G$ be an abelian group, and let $S \subseteq G$ be a pure subgroup. If $S \subseteq H \subseteq G$, prove that $H$ is a pure subgroup of $G$ if and only if $H / S$ is a pure subgroup of $G / S$.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator