Introduction to Abstract Algebra: with Applications

1.

Which of the following sets are rings with respect to the usual operations of addition and multiplication?

1. \(\displaystyle 7 {\mathbb Z} = \{7a : a \in {\mathbb Z}\}\)

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.

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 \({\mathbb Z}_2\)

3.

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

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

4.

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

5.

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

6.

Prove that

is a subring of \(\mathbb R\text{.}\)

7.

Prove or disprove: Any subring of a field \(F\) is an integral domain.

8.

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.

9.

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

10.

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

11.

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. \(1 \in S\text{.}\)

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

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

12.

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.

13.

Let \(R\) be a ring. Define the center of \(R\) to be

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

14.

Show that if \(R\) is any ring, then there is a unique homomorphism \(\phi \colon \mathbb Z \rightarrow R\text{.}\)

15.

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

16.

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

17.

Let \(\phi \colon R \rightarrow S\) 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. If \(R\) is a field, then \(\phi(R)\) is a field.

18.

Let \(p\) be prime. Prove that

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

19.

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

20.

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

21.

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.

22.

A field \(F\) is called a prime field if it has no proper subfields. If \(E\) is a subfield of \(F\) and \(E\) is a prime field, then \(E\) is a prime subfield of \(F\text{.}\)

1. Prove that every field contains a unique prime subfield.

2. If \(F\) is a field of characteristic 0, prove that the prime subfield of \(F\) is isomorphic to the field of rational numbers, \({\mathbb Q}\text{.}\)

3. If \(F\) is a field of characteristic \(p\text{,}\) prove that the prime subfield of \(F\) is isomorphic to \({\mathbb Z}_p\text{.}\)

23.

Let \(D\) be an integral domain.

1. Show that the operation of multiplication is well-defined in the field of fractions, \(F_D\text{.}\)

2. Verify the associative and commutative properties for addition in \(F_D\text{.}\)

3. Verify the associative and commutative properties for multiplication in \(F_D\text{.}\)

24.

Let \(R\) be a commutative ring with identity. We define a multiplicative subset of \(R\) to be a subset \(S\) such that \(1 \in S\) and \(ab \in S\) if \(a, b \in S\text{.}\)

1. Define a relation \(\sim\) on \(R \times S\) by \((a, s) \sim (a', s')\) if there exists an \(s^\ast \in S\) such that \(s^\ast(s' a -s a') = 0\text{.}\) Show that \(\sim\) is an equivalence relation on \(R \times S\text{.}\)

2. Let \(a/s\) denote the equivalence class of \((a,s) \in R \times S\) and let \(S^{-1}R\) be the set of all equivalence classes with respect to \(\sim\text{.}\) Define the operations of addition and multiplication on \(S^{-1} R\) by

   \begin{align*} \frac{a}{s} + \frac{b}{t} & = \frac{at + b s}{s t}\\ \frac{a}{s} \frac{b}{t} & = \frac{a b}{s t}, \end{align*}

   respectively. Prove that these operations are well-defined on \(S^{-1}R\) and that \(S^{-1}R\) is a ring with identity under these operations. The ring \(S^{-1}R\) is called the ring of quotients of \(R\) with respect to \(S\text{.}\)