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Section16.6Exercises

ΒΆ
"]}, {"cell_type":"markdown", "metadata":{}, "source":["
1

Which of the following sets are rings with respect to the usual operations of addition and multiplication? If the set is a ring, is it also a field?

  1. $7 {\\mathbb Z}$

  2. ${\\mathbb Z}_{18}$

  3. ${\\mathbb Q} ( \\sqrt{2}\\, ) = \\{a + b \\sqrt{2} : a, b \\in {\\mathbb Q}\\}$

  4. ${\\mathbb Q} ( \\sqrt{2}, \\sqrt{3}\\, ) = \\{a + b \\sqrt{2} + c \\sqrt{3} + d \\sqrt{6} : a, b, c, d \\in {\\mathbb Q}\\}$

  5. ${\\mathbb Z}[\\sqrt{3}\\, ] = \\{ a + b \\sqrt{3} : a, b \\in {\\mathbb Z} \\}$

  6. $R = \\{a + b \\sqrt[3]{3} : a, b \\in {\\mathbb Q} \\}$

  7. ${\\mathbb Z}[ i ] = \\{ a + b i : a, b \\in {\\mathbb Z} \\text{ and } i^2 = -1 \\}$

  8. ${\\mathbb Q}( \\sqrt[3]{3}\\, ) = \\{ a + b \\sqrt[3]{3} + c \\sqrt[3]{9} : a, b, c \\in {\\mathbb Q} \\}$

Hint

(a) $7 {\\mathbb Z}$ is a ring but not a field; (c) ${\\mathbb Q}(\\sqrt{2}\\, )$ is a field; (f) $R$ is not a ring.

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
2

Let $R$ be the ring of $2 \\times 2$ matrices of the form

\n\\begin{equation*}\n\\begin{pmatrix}\na & b \\\\\n0 & 0\n\\end{pmatrix},\n\\end{equation*}\n

where $a, b \\in {\\mathbb R}\\text{.}$ Show that although $R$ is a ring that has no identity, we can find a subring $S$ of $R$ with an identity.

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
3

List or characterize all of the units in each of the following rings.

  1. ${\\mathbb Z}_{10}$

  2. ${\\mathbb Z}_{12}$

  3. ${\\mathbb Z}_{7}$

  4. ${\\mathbb M}_2( {\\mathbb Z} )\\text{,}$ the $2 \\times 2$ matrices with entries in ${\\mathbb Z}$

  5. ${\\mathbb M}_2( {\\mathbb Z}_2 )\\text{,}$ the $2 \\times 2$ matrices with entries in ${\\mathbb Z}_2$

Hint

(a) $\\{1, 3, 7, 9 \\}\\text{;}$ (c) $\\{ 1, 2, 3, 4, 5, 6 \\}\\text{;}$ (e)

\n\\begin{equation*}\n\\left\\{\n\\begin{pmatrix}\n1 & 0 \\\\\n0 & 1\n\\end{pmatrix},\n\\begin{pmatrix}\n1 & 1 \\\\\n0 & 1\n\\end{pmatrix},\n\\begin{pmatrix}\n1 & 0 \\\\\n1 & 1\n\\end{pmatrix},\n\\begin{pmatrix}\n0 & 1 \\\\\n1 & 0\n\\end{pmatrix},\n\\begin{pmatrix}\n1 & 1 \\\\\n1 & 0\n\\end{pmatrix},\n\\begin{pmatrix}\n0 & 1 \\\\\n1 & 1\n\\end{pmatrix},\n\\right\\}.\n\\end{equation*}\n
"]}, {"cell_type":"markdown", "metadata":{}, "source":["
4

Find all of the ideals in each of the following rings. Which of these ideals are maximal and which are prime?

  1. ${\\mathbb Z}_{18}$

  2. ${\\mathbb Z}_{25}$

  3. ${\\mathbb M}_2( {\\mathbb R} )\\text{,}$ the $2 \\times 2$ matrices with entries in ${\\mathbb R}$

  4. ${\\mathbb M}_2( {\\mathbb Z} )\\text{,}$ the $2 \\times 2$ matrices with entries in ${\\mathbb Z}$

  5. ${\\mathbb Q}$

Hint

(a) $\\{0 \\}\\text{,}$ $\\{0, 9 \\}\\text{,}$ $\\{0, 6, 12 \\}\\text{,}$ $\\{0, 3, 6, 9, 12, 15 \\}\\text{,}$ $\\{0, 2, 4, 6, 8, 10, 12, 14, 16 \\}\\text{;}$ (c) there are no nontrivial ideals.

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
5

For each of the following rings $R$ with ideal $I\\text{,}$ give an addition table and a multiplication table for $R/I\\text{.}$

  1. $R = {\\mathbb Z}$ and $I = 6 {\\mathbb Z}$

  2. $R = {\\mathbb Z}_{12}$ and $I = \\{ 0, 3, 6, 9 \\}$

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
6

Find all homomorphisms $\\phi : {\\mathbb Z} / 6 {\\mathbb Z} \\rightarrow {\\mathbb Z} / 15 {\\mathbb Z}\\text{.}$

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
"]}, {"cell_type":"markdown", "metadata":{}, "source":["
7

Prove that ${\\mathbb R}$ is not isomorphic to ${\\mathbb C}\\text{.}$

Hint

Assume there is an isomorphism $\\phi: {\\mathbb C} \\rightarrow {\\mathbb R}$ with $\\phi(i) = a\\text{.}$

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
8

Prove or disprove: The ring ${\\mathbb Q}( \\sqrt{2}\\, ) = \\{ a + b \\sqrt{2} : a, b \\in {\\mathbb Q} \\}$ is isomorphic to the ring ${\\mathbb Q}( \\sqrt{3}\\, ) = \\{a + b \\sqrt{3} : a, b \\in {\\mathbb Q} \\}\\text{.}$

Hint

False. Assume there is an isomorphism $\\phi: {\\mathbb Q}(\\sqrt{2}\\, ) \\rightarrow {\\mathbb Q}(\\sqrt{3}\\, )$ such that $\\phi(\\sqrt{2}\\, ) = a\\text{.}$

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
9

What is the characteristic of the field formed by the set of matrices

\n\\begin{equation*}\nF =\n\\left\\{\n\\begin{pmatrix}\n1 & 0 \\\\\n0 & 1\n\\end{pmatrix},\n\\begin{pmatrix}\n1 & 1 \\\\\n1 & 0\n\\end{pmatrix},\n\\begin{pmatrix}\n0 & 1 \\\\\n1 & 1\n\\end{pmatrix},\n\\begin{pmatrix}\n0 & 0 \\\\\n0 & 0\n\\end{pmatrix}\n\\right\\}\n\\end{equation*}\n

with entries in ${\\mathbb Z}_2\\text{?}$

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
10

Define a map $\\phi : {\\mathbb C} \\rightarrow {\\mathbb M}_2 ({\\mathbb R})$ by

\n\\begin{equation*}\n\\phi( a + bi) =\n\\begin{pmatrix}\na & b \\\\\n-b & a\n\\end{pmatrix}.\n\\end{equation*}\n

Show that $\\phi$ is an isomorphism of ${\\mathbb C}$ with its image in ${\\mathbb M}_2 ({\\mathbb R})\\text{.}$

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
"]}, {"cell_type":"markdown", "metadata":{}, "source":["
11

Prove that the Gaussian integers, ${\\mathbb Z}[i ]\\text{,}$ are an integral domain.

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
12

Prove that ${\\mathbb Z}[ \\sqrt{3}\\, i ] = \\{ a + b \\sqrt{3}\\, i : a, b \\in {\\mathbb Z} \\}$ is an integral domain.

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
13

Solve each of the following systems of congruences.

  1. \n\\begin{align*}\nx & \\equiv 2 \\pmod{5}\\\\\nx & \\equiv 6 \\pmod{11}\n\\end{align*}\n
  2. \n\\begin{align*}\nx & \\equiv 3 \\pmod{7}\\\\\nx & \\equiv 0 \\pmod{8}\\\\\nx & \\equiv 5 \\pmod{15}\n\\end{align*}\n
  3. \n\\begin{align*}\nx & \\equiv 2 \\pmod{4}\\\\\nx & \\equiv 4 \\pmod{7}\\\\\nx & \\equiv 7 \\pmod{9}\\\\\nx & \\equiv 5 \\pmod{11}\n\\end{align*}\n
  4. \n\\begin{align*}\nx & \\equiv 3 \\pmod{5}\\\\\nx & \\equiv 0 \\pmod{8}\\\\\nx & \\equiv 1 \\pmod{11}\\\\\nx & \\equiv 5 \\pmod{13}\n\\end{align*}\n
Hint

(a) $x \\equiv 17 \\pmod{55}\\text{;}$ (c) $x \\equiv 214 \\pmod{2772}\\text{.}$

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
14

Use the method of parallel computation outlined in the text to calculate $2234 + 4121$ by dividing the calculation into four separate additions modulo 95, 97, 98, and 99.

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
"]}, {"cell_type":"markdown", "metadata":{}, "source":["
15

Explain why the method of parallel computation outlined in the text fails for $2134 \\cdot 1531$ if we attempt to break the calculation down into two smaller calculations modulo 98 and 99.

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
16

If $R$ is a field, show that the only two ideals of $R$ are $\\{ 0 \\}$ and $R$ itself.

Hint

If $I \\neq \\{ 0 \\}\\text{,}$ show that $1 \\in I\\text{.}$

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
17

Let $a$ be any element in a ring $R$ with identity. Show that $(-1)a = -a\\text{.}$

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
18

Let $\\phi : R \\rightarrow S$ be a ring homomorphism. Prove each of the following statements.

  1. If $R$ is a commutative ring, then $\\phi(R)$ is a commutative ring.

  2. $\\phi( 0 ) = 0\\text{.}$

  3. Let $1_R$ and $1_S$ be the identities for $R$ and $S\\text{,}$ respectively. If $\\phi$ is onto, then $\\phi(1_R) = 1_S\\text{.}$

  4. If $R$ is a field and $\\phi(R) \\neq 0\\text{,}$ then $\\phi(R)$ is a field.

Hint

(a) $\\phi(a) \\phi(b) = \\phi(ab) = \\phi(ba) = \\phi(b) \\phi(a)\\text{.}$

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
19

Prove that the associative law for multiplication and the distributive laws hold in $R/I\\text{.}$

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
20

Prove the Second Isomorphism Theorem for rings: Let $I$ be a subring of a ring $R$ and $J$ an ideal in $R\\text{.}$ Then $I \\cap J$ is an ideal in $I$ and

\n\\begin{equation*}\nI / I \\cap J \\cong I + J /J.\n\\end{equation*}\n
"]}, {"cell_type":"markdown", "metadata":{}, "source":["
"]}, {"cell_type":"markdown", "metadata":{}, "source":["
21

Prove the Third Isomorphism Theorem for rings: Let $R$ be a ring and $I$ and $J$ be ideals of $R\\text{,}$ where $J \\subset I\\text{.}$ Then

\n\\begin{equation*}\nR/I \\cong \\frac{R/J}{I/J}.\n\\end{equation*}\n
"]}, {"cell_type":"markdown", "metadata":{}, "source":["
"]}, {"cell_type":"markdown", "metadata":{}, "source":["
22

Prove the Correspondence Theorem: Let $I$ be an ideal of a ring $R\\text{.}$ Then $S \\rightarrow S/I$ is a one-to-one correspondence between the set of subrings $S$ containing $I$ and the set of subrings of $R/I\\text{.}$ Furthermore, the ideals of $R$ correspond to ideals of $R/I\\text{.}$

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
"]}, {"cell_type":"markdown", "metadata":{}, "source":["
23

Let $R$ be a ring and $S$ a subset of $R\\text{.}$ Show that $S$ is a subring of $R$ if and only if each of the following conditions is satisfied.

  1. $S \\neq \\emptyset\\text{.}$

  2. $rs \\in S$ for all $r, s \\in S\\text{.}$

  3. $r - s \\in S$ for all $r, s \\in S\\text{.}$

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
24

Let $R$ be a ring with a collection of subrings $\\{ R_{\\alpha} \\}\\text{.}$ Prove that $\\bigcap R_{\\alpha}$ is a subring of $R\\text{.}$ Give an example to show that the union of two subrings is not necessarily a subring.

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
25

Let $\\{ I_{\\alpha} \\}_{\\alpha \\in A}$ be a collection of ideals in a ring $R\\text{.}$ Prove that $\\bigcap_{\\alpha \\in A} I_{\\alpha}$ is also an ideal in $R\\text{.}$ Give an example to show that if $I_1$ and $I_2$ are ideals in $R\\text{,}$ then $I_1 \\cup I_2$ may not be an ideal.

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
26

Let $R$ be an integral domain. Show that if the only ideals in $R$ are $\\{ 0 \\}$ and $R$ itself, $R$ must be a field.

Hint

Let $a \\in R$ with $a \\neq 0\\text{.}$ Then the principal ideal generated by $a$ is $R\\text{.}$ Thus, there exists a $b \\in R$ such that $ab =1\\text{.}$

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
27

Let $R$ be a commutative ring. An element $a$ in $R$ is nilpotent if $a^n = 0$ for some positive integer $n\\text{.}$ Show that the set of all nilpotent elements forms an ideal in $R\\text{.}$

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
28

A ring $R$ is a Boolean ring if for every $a \\in R\\text{,}$ $a^2 = a\\text{.}$ Show that every Boolean ring is a commutative ring.

Hint

Compute $(a+b)^2$ and $(-ab)^2\\text{.}$

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
"]}, {"cell_type":"markdown", "metadata":{}, "source":["
29

Let $R$ be a ring, where $a^3 =a$ for all $a \\in R\\text{.}$ Prove that $R$ must be a commutative ring.

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
"]}, {"cell_type":"markdown", "metadata":{}, "source":["
30

Let $R$ be a ring with identity $1_R$ and $S$ a subring of $R$ with identity $1_S\\text{.}$ Prove or disprove that $1_R = 1_S\\text{.}$

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
"]}, {"cell_type":"markdown", "metadata":{}, "source":["
31

If we do not require the identity of a ring to be distinct from 0, we will not have a very interesting mathematical structure. Let $R$ be a ring such that $1 = 0\\text{.}$ Prove that $R = \\{ 0 \\}\\text{.}$

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
32

Let $S$ be a nonempty subset of a ring $R\\text{.}$ Prove that there is a subring $R'$ of $R$ that contains $S\\text{.}$

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
33

Let $R$ be a ring. Define the center of $R$ to be

\n\\begin{equation*}\nZ(R) = \\{ a \\in R : ar = ra \\text{ for all } r \\in R \\}.\n\\end{equation*}\n

Prove that $Z(R)$ is a commutative subring of $R\\text{.}$

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
34

Let $p$ be prime. Prove that

\n\\begin{equation*}\n{\\mathbb Z}_{(p)} = \\{ a / b : a, b \\in {\\mathbb Z} \\text{ and } \\gcd( b,p) = 1 \\} \n\\end{equation*}\n

is a ring. The ring ${\\mathbb Z}_{(p)}$ is called the ring of integers localized at $p\\text{.}$

Hint

Let $a/b, c/d \\in {\\mathbb Z}_{(p)}\\text{.}$ Then $a/b + c/d = (ad + bc)/bd$ and $(a/b) \\cdot (c/d) = (ac)/(bd)$ are both in ${\\mathbb Z}_{(p)}\\text{,}$ since $\\gcd(bd,p) = 1\\text{.}$

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
"]}, {"cell_type":"markdown", "metadata":{}, "source":["
35

Prove or disprove: Every finite integral domain is isomorphic to ${\\mathbb Z}_p\\text{.}$

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
"]}, {"cell_type":"markdown", "metadata":{}, "source":["
36

Let $R$ be a ring with identity.

  1. Let $u$ be a unit in $R\\text{.}$ Define a map $i_u : R \\rightarrow R$ by $r \\mapsto uru^{-1}\\text{.}$ Prove that $i_u$ is an automorphism of $R\\text{.}$ Such an automorphism of $R$ is called an inner automorphism of $R\\text{.}$ Denote the set of all inner automorphisms of $R$ by $\\inn(R)\\text{.}$

  2. Denote the set of all automorphisms of $R$ by $\\aut(R)\\text{.}$ Prove that $\\inn(R)$ is a normal subgroup of $\\aut(R)\\text{.}$

  3. Let $U(R)$ be the group of units in $R\\text{.}$ Prove that the map

    \n\\begin{equation*}\n\\phi : U(R) \\rightarrow \\inn(R)\n\\end{equation*}\n

    defined by $u \\mapsto i_u$ is a homomorphism. Determine the kernel of $\\phi\\text{.}$

  4. Compute $\\aut( {\\mathbb Z})\\text{,}$ $\\inn( {\\mathbb Z})\\text{,}$ and $U( {\\mathbb Z})\\text{.}$

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
37

Let $R$ and $S$ be arbitrary rings. Show that their Cartesian product is a ring if we define addition and multiplication in $R \\times S$ by

  1. $(r, s) + (r', s') = ( r + r', s + s')$

  2. $(r, s)(r', s') = ( rr', ss')$

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
"]}, {"cell_type":"markdown", "metadata":{}, "source":["
38

An element $x$ in a ring is called an idempotent if $x^2 = x\\text{.}$ Prove that the only idempotents in an integral domain are $0$ and $1\\text{.}$ Find a ring with a idempotent $x$ not equal to 0 or 1.

Hint

Suppose that $x^2 = x$ and $x \\neq 0\\text{.}$ Since $R$ is an integral domain, $x = 1\\text{.}$ To find a nontrivial idempotent, look in ${\\mathbb M}_2({\\mathbb R})\\text{.}$

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
"]}, {"cell_type":"markdown", "metadata":{}, "source":["
39

Let $\\gcd(a, n) = d$ and $\\gcd(b, d) \\neq 1\\text{.}$ Prove that $ax \\equiv b \\pmod{n}$ does not have a solution.

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
"]}, {"cell_type":"markdown", "metadata":{}, "source":["
40The Chinese Remainder Theorem for Rings

Let $R$ be a ring and $I$ and $J$ be ideals in $R$ such that $I+J = R\\text{.}$

  1. Show that for any $r$ and $s$ in $R\\text{,}$ the system of equations

    \n\\begin{align*}\nx & \\equiv r \\pmod{I}\\\\\nx & \\equiv s \\pmod{J}\n\\end{align*}\n

    has a solution.

  2. In addition, prove that any two solutions of the system are congruent modulo $I \\cap J\\text{.}$

  3. Let $I$ and $J$ be ideals in a ring $R$ such that $I + J = R\\text{.}$ Show that there exists a ring isomorphism

    \n\\begin{equation*}\nR/(I \\cap J) \\cong R/I \\times R/J.\n\\end{equation*}\n
"]}, {"cell_type":"markdown", "metadata":{}, "source":["
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