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

2

Let $R$ be the ring of $2 \times 2$ matrices of the form \begin{equation*}\begin{pmatrix} a & b \\ 0 & 0 \end{pmatrix},\end{equation*} where $a, b \in {\mathbb R}$. 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. ${\mathbb Z}_{10}$

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

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

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

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

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

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

5. ${\mathbb Q}$

5

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

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

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

6

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

7

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

8

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

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 ${\mathbb Z}_2$?

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

11

Prove that the Gaussian integers, ${\mathbb Z}[i ]$, 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*}

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.

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.

16

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

17

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

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. $\phi( 0 ) = 0$.

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

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

19

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

20

Prove the Second Isomorphism Theorem for rings: Let $I$ be a subring of a ring $R$ and $J$ an ideal in $R$. Then $I \cap J$ is an ideal in $I$ and \begin{equation*}I / I \cap J \cong I + J /J.\end{equation*}

21

Prove the Third Isomorphism Theorem for rings: Let $R$ be a ring and $I$ and $J$ be ideals of $R$, where $J \subset I$. Then \begin{equation*}R/I \cong \frac{R/J}{I/J}.\end{equation*}

22

Prove the Correspondence Theorem: Let $I$ be an ideal of a ring $R$. 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 $R/I$. Furthermore, the ideals of $R$ correspond to ideals of $R/I$.

23

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

1. $S \neq \emptyset$.

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

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

24

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

25

Let $\{ I_{\alpha} \}_{\alpha \in A}$ be a collection of ideals in a ring $R$. Prove that $\bigcap_{\alpha \in A} I_{\alpha}$ is also an ideal in $R$. Give an example to show that if $I_1$ and $I_2$ are ideals in $R$, 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.

27

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

28

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

29

Let $R$ be a ring, where $a^3 =a$ for all $a \in R$. 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 $1_S$. Prove or disprove that $1_R = 1_S$.

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 $1 = 0$. Prove that $R = \{ 0 \}$.

32

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

33

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

34

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 $p$.

35

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

36

Let $R$ be a ring with identity.

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

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

3. Let $U(R)$ be the group of units in $R$. 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 $\phi$.

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

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

38

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

39

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

40The Chinese Remainder Theorem for Rings

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

1. Show that for any $r$ and $s$ in $R$, 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 $I \cap J$.

3. Let $I$ and $J$ be ideals in a ring $R$ such that $I + J = R$. Show that there exists a ring isomorphism \begin{equation*}R/(I \cap J) \cong R/I \times R/J.\end{equation*}