Abstract Algebra: Theory and Applications

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

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

with entries in \({\mathbb Z}_2\text{?}\)

10.

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

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

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

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

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

33.

Let \(p\) be prime. Prove that

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*}