## Exercises16.7Exercises

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

###### 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 $${\mathbb Z}_2$$

###### 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. $$\displaystyle {\mathbb Q}$$

###### 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 $${\mathbb C}\text{.}$$

###### 8.

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

###### 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*}
###### 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.

###### 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.

###### 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.

###### 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.

###### 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 $$p\text{.}$$

###### 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.

###### 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*}