###### 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. $\displaystyle 7 {\mathbb Z}$

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

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

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

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

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

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

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

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.

###### 2.

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

\begin{equation*} \begin{pmatrix} a & b \\ 0 & 0 \end{pmatrix}\text{,} \end{equation*}

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.

###### 3.

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

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

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

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

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)

\begin{equation*} \left\{ \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}, \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}, \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 1 \\ 1 & 1 \end{pmatrix}, \right\}\text{.} \end{equation*}
###### 4.

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

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

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

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. $\displaystyle {\mathbb Q}$

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.

###### 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 $I = \{ 0, 3, 6, 9 \}$

###### 6.

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

###### 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 $\phi(i) = a\text{.}$

###### 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 $\phi(\sqrt{2}\, ) = a\text{.}$

###### 9.

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

\begin{equation*} F = \left\{ \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 1 \\ 1 & 1 \end{pmatrix}, \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} \right\} \end{equation*}

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

###### 10.

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

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

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

###### 11.

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

###### 12.

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

###### 13.

Solve each of the following systems of congruences.

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

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

###### 14.

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

###### 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 $99\text{.}$

###### 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 $1 \in I\text{.}$

###### 17.

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

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

###### 19.

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

###### 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

\begin{equation*} I / I \cap J \cong I + J /J\text{.} \end{equation*}
###### 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

\begin{equation*} R/I \cong \frac{R/J}{I/J}\text{.} \end{equation*}
###### 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 $R/I\text{.}$

###### 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 $r, s \in S\text{.}$

###### 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.

###### 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.

###### 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 $ab =1\text{.}$

###### 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 $R\text{.}$

###### 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 $(-ab)^2\text{.}$

###### 29.

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

###### 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 $1_R = 1_S\text{.}$

###### 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 $R = \{ 0 \}\text{.}$

###### 32.

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

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

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

###### 33.

Let $p$ be prime. Prove that

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

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 $\gcd(bd,p) = 1\text{.}$

###### 34.

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

###### 35.

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

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

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 $U( {\mathbb Z})\text{.}$

###### 36.

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. $\displaystyle (r, s) + (r', s') = ( r + r', s + s')$

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

###### 37.

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 ${\mathbb M}_2({\mathbb R})\text{.}$

###### 38.

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

###### 39.The 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

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

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

\begin{equation*} R/(I \cap J) \cong R/I \times R/J\text{.} \end{equation*}