Contemporary Abstract Algebra

Joseph Gallian

Chapter 14

Ideals and Factor Rings - all with Video Answers

Educators

Chapter Questions

Problem 1

Verify that the set defined in Example 3 is an ideal.

Adriano Chikande

Adriano Chikande

Numerade Educator

Problem 2

Verify that the set $A$ in Example 4 is an ideal and that $A=\langle x\rangle$.

Tanishq Gupta

Numerade Educator

Problem 3

Verify that the set $I$ in Example 5 is an ideal and that if $J$ is any ideal of $R$ that contains $a_{1}, a_{2}, \ldots, a_{n}$, then $I \subseteq J .$ (Hence, $\left\langle a_{1}\right.$, $\left.a_{2}, \ldots, a_{n}\right\rangle$ is the smallest ideal of $R$ that contains $a_{1}, a_{2}, \ldots, a_{n n} .$

Prathan Jarupoonphol

Prathan Jarupoonphol

Numerade Educator

Problem 4

Find a subring of $Z \oplus Z$ that is not an ideal of $Z \oplus Z$.

Uma Kumari

Numerade Educator

Problem 5

Let $S=\{a+b i \mid a, b \in Z, b$ is even $\} .$ Show that $S$ is a subring of $Z[i]$, but not an ideal of $Z[i]$.

Avinash Vishwakarma

Avinash Vishwakarma

Numerade Educator

Problem 6

Find all maximal ideals in
a. $Z_{8}$.
b. $Z_{10^{\circ}}$
c. $Z_{12}$
d. $Z_{n^{*}}$

Lily An

Numerade Educator

Problem 7

Let $a$ belong to a commutative ring $R$. Show that $a R=\{a r \mid r \in R\}$ is an ideal of $R$. If $R$ is the ring of even integers, list the elements of $4 R$.

Abigail Martyr

Numerade Educator

Problem 8

Prove that the intersection of any set of ideals of a ring is an ideal.

Mengchun Cai

Numerade Educator

Problem 9

If $n$ is an integer greater than 1 , show that $\langle n\rangle=n Z$ is a prime ideal of $Z$ if and only if $n$ is prime. (This exercise is referred to in this chapter.)

Chris Trentman

Numerade Educator

Problem 10

If $A$ and $B$ are ideals of a ring, show that the $s u m$ of $A$ and $B, A+B=$ $\{a+b \mid a \in A, b \in B\}$, is an ideal.

Mohan Jain

Numerade Educator

Problem 11

In the ring of integers, find a positive integer $a$ such that
a. $\langle a\rangle=\langle 2\rangle+\langle 3\rangle$.
b. $\langle a\rangle=\langle 6\rangle+\langle 8\rangle$.
c. $\langle a\rangle=\langle m\rangle+\langle n\rangle$.

Gaurav Kalra

Numerade Educator

Problem 12

If $A$ and $B$ are ideals of a ring, show that the product of $A$ and $B$, $A B=\left\{a_{1} b_{1}+a_{2} b_{2}+\cdots+a_{n} b_{n} \mid a_{i} \in A, b_{i} \in B, n\right.$ a positive
integer $\}$, is an ideal.

Chris Trentman

Numerade Educator

Problem 13

Find a positive integer $a$ such that
a. $\langle a\rangle=\langle 3\rangle\langle 4\rangle$.
b. $\langle a\rangle=\langle 6\rangle(8\rangle$.
c. $\langle a\rangle=\langle m\rangle\langle n\rangle$.

Mengchun Cai

Numerade Educator

Problem 14

Let $A$ and $B$ be ideals of a ring. Prove that $A B \subseteq A \cap B$.

WM

William Mead

Numerade Educator

Problem 15

If $A$ is an ideal of a ring $R$ and 1 belongs to $A$, prove that $A=R$. (This exercise is referred to in this chapter.)

Chris Trentman

Numerade Educator

Problem 16

If $A$ and $B$ are ideals of a commutative ring $R$ with unity and $A+B=R$, show that $A \cap B=A B$.

Paul A.

California State Polytechnic University, Pomona

Problem 17

If an ideal $I$ of a ring $R$ contains a unit, show that $I=R$.

Angelo Rendina

Numerade Educator

Problem 18

Suppose that in the ring $Z$, the ideal $\langle 35\rangle$ is a proper ideal of $J$ and $J$ is a proper ideal of $1 .$ What are the possibilities for $J ?$ What are the possibilities for $I ?$

Narayan Hari

Numerade Educator

Problem 19

Give an example of a ring that has exactly two maximal ideals.

Diwakar Mandilwar

Diwakar Mandilwar

Numerade Educator

Problem 20

Suppose that $R$ is a commutative ring and $|R|=30 .$ If $I$ is an ideal of $R$ and $|R|=10$, prove that $I$ is a maximal ideal.

Prathan Jarupoonphol

Prathan Jarupoonphol

Numerade Educator

Problem 21

Let $R$ and $I$ be as described in Example $10 .$ Prove that $I$ is an ideal of $R$

Julie Silva

Numerade Educator

Problem 22

Let $I=\langle 2\rangle .$ Prove that $I[x]$ is not a maximal ideal of $Z[x]$ even though $I$ is a maximal ideal of $Z$.

Adrian Co

Numerade Educator

Problem 23

Verify the claim made in Example 10 about the size of $R / I$.

Matthew Dunham

Numerade Educator

Problem 24

Give an example of a commutative ring that has a maximal ideal that is not a prime ideal.

Amy Jiang

Numerade Educator

Problem 25

Show that the set $B$ in the latter half of the proof of Theorem $14.4$ is an ideal of $R .$ (This exercise is referred to in this chapter.)

Raj Bala

Numerade Educator

Problem 26

If $R$ is a commutative ring with unity and $A$ is a proper ideal of $R$, show that $R / A$ is a commutative ring with unity.

J Alessandro Briseño-Tapia

J Alessandro Briseño-Tapia

Numerade Educator

Problem 27

Prove that the only ideals of a field $F$ are $\{0\}$ and $F$ itself.

Lucas Finney

Numerade Educator

Problem 28

Show that $\mathbf{R}[x] /\left\langle x^{2}+1\right\rangle$ is a field.

Bryan Lynn

Numerade Educator

Problem 29

In $Z[x]$, the ring of polynomials with integer coefficients, let $I=$ $\{f(x) \in Z[x] \mid f(0)=0\}$. Prove that $I=\langle x\rangle$. (This exercise is referred to in this chapter and in Chapter $15 .$ )

Ahmad Reda

Numerade Educator

Problem 30

Show that $A=\{(3 x, y) \mid x, y \in Z\}$ is a maximal ideal of $Z \oplus Z$. Generalize. What happens if $3 x$ is replaced by $4 x$ ? Generalize.

Uma Kumari

Numerade Educator

Problem 31

Let $R$ be the ring of continuous functions from $\mathbf{R}$ to $\mathbf{R}$. Show that $A=\{f \in R \mid f(0)=0\}$ is a maximal ideal of $R$

Nick Johnson

Numerade Educator

Problem 32

Let $R=Z_{8} \oplus Z_{30} .$ Find all maximal ideals of $R$, and for each maximal ideal $I$, identify the size of the field $R / I$.

Lucas Finney

Numerade Educator

Problem 33

How many elements are in $Z[i] /\langle 3+i\rangle ?$ Give reasons for your answer.

Shahina -

Numerade Educator

Problem 34

In $Z[x]$, the ring of polynomials with integer coefficients, let $I=$ $\{f(x) \in Z[x] \mid f(0)=0\} .$ Prove that $I$ is not a maximal ideal.

Hast Aggarwal

Numerade Educator

Problem 35

In $Z \oplus Z$, let $I=\{(a, 0) \mid a \in Z\} .$ Show that $I$ is a prime ideal but not a maximal ideal.

Avinash Vishwakarma

Avinash Vishwakarma

Numerade Educator

Problem 36

Let $R$ be a ring and let $I$ be an ideal of $R$. Prove that the factor ring $R / I$ is commutative if and only if $r s-s r \in I$ for all $r$ and $s$ in $R$.

Sriparna Bhattacharjee

Sriparna Bhattacharjee

Numerade Educator

Problem 37

In $Z[x]$, let $I=|f(x) \in Z[x]| f(0)$ is an even integer $\}$. Prove that $I=\langle x, 2\rangle .$ Is $I$ a prime ideal of $Z[x] ?$ Is $I$ a maximal ideal? How many elements does $Z[x] / I$ have? (This exercise is referred to in this chapter.)

WZ

Wen Zheng

Numerade Educator

Problem 38

Prove that $I=\langle 2+2 i\rangle$ is not a prime ideal of $Z[i] .$ How many elements are in $Z[i] / I ?$ What is the characteristic of $Z[i] / I ?$

AG

Ankit Gupta

Numerade Educator

Problem 39

$\operatorname{In} Z_{5}[x]$, let $I=\left\langle x^{2}+x+2\right\rangle .$ Find the multiplicative inverse of $2 x+$ $3+I$ in $Z_{5}[x] / I$

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 40

Let $R$ be a ring and let $p$ be a fixed prime. Show that $I_{p}=\{r \in R$ I additive order of $r$ is a power of $p\}$ is an ideal of $R$.

Sriparna Bhattacharjee

Sriparna Bhattacharjee

Numerade Educator

Problem 41

An integral domain $D$ is called a principal ideal domain if every ideal of $D$ has the form $\langle a\rangle=\{a d \mid d \in D\}$ for some $a$ in $D$. Show that $Z$ is a principal ideal domain. (This exercise is referred to in Chapter 18.)

Adrian Co

Numerade Educator

Problem 42

Let $R=\left\{\left[\begin{array}{ll}a & b \\ 0 & d\end{array}\right] \mid a, b, d \in Z\right\}$ and $S=\left\{\left[\begin{array}{cc}r & s \\ 0 & t\end{array}\right] \mid r, s, t \in Z, s\right.$
is even\}. If $S$ is an ideal of $R$, what can you say about $r$ and $t$ ?

Gaurav Kalra

Numerade Educator

Problem 43

If $R$ and $S$ are principal ideal domains, prove that $R \oplus S$ is a principal ideal ring. (See Exercise 41 for the definition.)

HN

Harrison Nascimento

Numerade Educator

Problem 44

Let $a$ and $b$ belong to a commutative ring $R$. Prove that $\{x \in R$ ? $a x \in b R\}$ is an ideal.

Paul A.

California State Polytechnic University, Pomona

Problem 45

Let $R$ be a commutative ring and let $A$ be any subset of $R$. Show that the annihilator of $A, \operatorname{Ann}(A)=\{r \in R \mid r a=0$ for all $a \operatorname{in} A\}$, is an ideal.

Vishvajeetkumar Bhaskar Batule

Vishvajeetkumar Bhaskar Batule

Numerade Educator

Problem 46

Let $R$ be a commutative ring and let $A$ be any ideal of $R$. Show that the nil radical of $A, N(A)=\left\{r \in R \mid r^{n} \in A\right.$ for some positive integer $n$ ( $n$ depends on $r$ ) $\}$, is an ideal of $R$. $[N(\langle 0\rangle)$ is called the nil radical of $R .]$

Amy Jiang

Numerade Educator

Problem 47

Let $R=Z_{27}$. Find
a. $N(\langle 0\rangle)$.
b. $N((3\rangle)$.
c. $N(\langle 9\rangle)$.

Gaurav Kalra

Numerade Educator

Problem 48

Let $R=Z_{36}$. Find
a. $N(\langle 0\rangle)$.
b. $N(\langle 4\rangle)$.
c. $N(\langle 6\rangle)$.

Aman Gupta

Numerade Educator

Problem 49

Let $R$ be a commutative ring. Show that $R / N(\langle 0)$ ) has no nonzero nilpotent elements.

Joe Lesueur

Numerade Educator

Problem 50

Let $A$ be an ideal of a commutative ring. Prove that $N(N(A))=N(A)$.

Aayush Gupta

Numerade Educator

Problem 51

Let $Z_{2}[x]$ be the ring of all polynomials with coefficients in $Z_{2}$ (that is, coefficients are 0 or 1 , and addition and multiplication of coefficients are done modulo 2). Show that $Z_{2}[x] /\left\langle x^{2}+x+1\right\rangle$ is a field.

Ahmad Reda

Numerade Educator

Problem 52

List the elements of the field given in Exercise 51 , and make an addition and multiplication table for the field.

Victor Salazar

Numerade Educator

Problem 53

Show that $Z_{3}[x] /\left\langle x^{2}+x+1\right\rangle$ is not a field.

Harshita Goel

Numerade Educator

Problem 54

Let $R$ be a commutative ring without unity, and let $a \in R$. Describe the smallest ideal $I$ of $R$ that contains $a$ (that is, if $J$ is any ideal that contains $a$, then $I \subseteq J$ ).

Bobby Barnes

University of North Texas

Problem 55

Let $R$ be the ring of continuous functions from $\mathbf{R}$ to $\mathbf{R}$. Let $A=$ $\{f \in R \mid f(0)$ is an even integer\}. Show that $A$ is a subring of $R$,
but not an ideal of $R$.

Srilakshmi E K

Numerade Educator

Problem 56

Show that $Z[i] /\langle 1-i\rangle$ is a field. How many elements does this field have?

Chris Trentman

Numerade Educator

Problem 57

If $R$ is a principal ideal domain and $I$ is an ideal of $R$, prove that every ideal of $R / I$ is principal (see Exercise 41$)$.

Himanshu Kushwaha

Himanshu Kushwaha

Numerade Educator

Problem 58

How many elements are in $Z_{5}[i] /\langle 1+i\rangle ?$

Himanshu Kushwaha

Himanshu Kushwaha

Numerade Educator

Problem 59

Let $R$ be a commutative ring with unity that has the property that $a^{2}=a$ for all $a$ in $R$. Let $I$ be a prime ideal in $R$. Show that $\mathbb{R} / I=2$.

Chandra Jain

Numerade Educator

Problem 60

Let $R$ be a commutative ring with unity, and let $I$ be a proper ideal with the property that every element of $R$ that is not in $I$ is a unit of $R$. Prove that $I$ is the unique maximal ideal of $R$.

Prathan Jarupoonphol

Prathan Jarupoonphol

Numerade Educator

Problem 61

Let $I_{0}=\{f(x) \in Z[x] \mid f(0)=0\}$. For any positive integer $n$, show that there exists a sequence of strictly increasing ideals such that $I_{0} \subset I_{1} \subset I_{2} \subset \cdots \subset I_{n} \subset Z[x]$

Prathan Jarupoonphol

Prathan Jarupoonphol

Numerade Educator

Problem 62

Let $R=\left[\left(a_{1}, a_{2}, a_{3}, \ldots\right)\right\}$, where each $a_{i} \in Z$. Let $I=\left\{\left(a_{1}, a_{2},\right.\right.$,
$a_{3}, \ldots$ ) $\}$, where only a finite number of terms are nonzero. Prove that $I$ is not a principal ideal of $R$.

Ryan Swift

Numerade Educator

Problem 63

Let $R$ be a commutative ring with unity and let $a, b \in R .$ Show that $\langle a, b\rangle$, the smallest ideal of $R$ containing $a$ and $b$, is $I=\{r a+s b \mid$ $r, s \in R\} .$ That is, show that $I$ contains $a$ and $b$ and that any ideal that contains $a$ and $b$ also contains $I$.

Bobby Barnes

University of North Texas