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Exercises 4.5 Exercises

1.

Which of the following subsets of \(\mathbb C\) are subrings?
  1. \(\displaystyle 7 {\mathbb Z} = \{7a : a \in {\mathbb Z}\}\)
  2. \(\displaystyle {\mathbb Q} ( \sqrt{2}\, ) = \{a + b \sqrt{2} : a, b \in {\mathbb Q}\}\)
  3. \(\displaystyle {\mathbb Q} ( \sqrt{2}, \sqrt{3}\, ) = \{a + b \sqrt{2} + c \sqrt{3} + d \sqrt{6} : a, b, c, d \in {\mathbb Q}\}\)
  4. \(\displaystyle {\mathbb Z}[\sqrt{3}\, ] = \{ a + b \sqrt{3} : a, b \in {\mathbb Z} \}\)
  5. \(\displaystyle R = \{a + b \sqrt[3]{3} : a, b \in {\mathbb Q} \}\)
  6. \(\displaystyle {\mathbb Z}[ i ] = \{ a + b i : a, b \in {\mathbb Z} \text{ and } i^2 = -1 \}\)
  7. \(\displaystyle {\mathbb Q}( \sqrt[3]{3}\, ) = \{ a + b \sqrt[3]{3} + c \sqrt[3]{9} : a, b, c \in {\mathbb Q} \}\)

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.

Consider 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 \({\mathbb Z}_2\text{?}\)
  1. Verify that \(F\) is a subring of the ring of matrices with entries in \(\mathbb Z_2\) and make an addition table and multiplication table for \(F\text{.}\)
  2. What is the characteristic of \(F\text{?}\)
  3. Is \(F\) commutative?
  4. Is \(F\) a field?

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
\begin{equation*} {\mathbb Z}\left[\textstyle\frac{1+\sqrt{5}}{2}\right] = \left\{a + b \frac{1+\sqrt{5}}{2} : a, b \in \mathbb Z\right\} \end{equation*}
is a subring of \(\mathbb R\text{.}\)

7.

Let \(S\) be a non-empty set, and as usual we use \(\mathcal P(S)\) to denote the power set of \(S\text{,}\) which is the set of all subsets of \(S\text{.}\) In this exercise, we let \(R\) be the set \(\mathcal P(S)\) together with the following two operations on \(\mathcal P(S)\text{:}\)
\begin{align*} A + B &= A \cup B \setminus (A \cap B)\\ A \cdot B &= A \cap B \end{align*}
We will show that these operations define a commutative ring on \(R\text{.}\)
  1. Show that addition is commutative: \(A+B = B+A\) for all \(A,B \in R\text{.}\)
  2. Show that addition is associative: \((A+B)+C = A+(B+C)\) for all \(A,B,C \in R\text{.}\)
  3. Show that there is an element \(0 \in R\) such that \(A+0=A\) for all \(A \in R\text{.}\)
  4. Show that \(R\) has inverses: For any element \(A \in R\text{,}\) there exists an element \(-A\in R\) such that \(A+(-A) = 0\text{.}\)
  5. Show that multiplication is associative: \((A\cdot B)\cdot C = A \cdot (B \cdot C)\) for all \(A,B,C \in R\text{.}\)
  6. Show that multiplication is commutative: \(A \cdot B=B \cdot A\) for all \(A,B \in R\text{.}\)
  7. Show that there is an element \(1 \in R\) such that \(1\cdot A = A\cdot 1 = A\) for all \(A \in R\text{.}\)
  8. Show that \(A \cdot (B+C)=A\cdot B + A \cdot C\) for all \(A,B,C \in R\text{.}\)
  9. What is the characteristic of \(R\text{?}\)

8.

Let \(R\) be a ring. We use \(R[i]\) to denote the set of pairs \((a,b) \in R \times R\text{,}\) with the following operations:
\begin{gather*} (a,b) + (a',b') = (a+a', b+b')\\ (a,b)\cdot (a',b') = (aa'-bb',ab'+ba') \end{gather*}
Notice that these operations are similar to the definition of complex numbers as pairs of real numbers. This exercise will show that \(R[i]\) is a ring.
  1. Show that addition is commutative: \(\alpha+\beta = \beta+a\alpha\) for all \(\alpha,\beta \in R[i]\text{.}\)
  2. Show that addition is associative: \((\alpha+\beta)+\gamma = \alpha+(\beta+\gamma)\) for all \(\alpha,\beta \in R[i]\text{.}\)
  3. Show that there is an element \(0 \in R[i]\) such that \(\alpha + 0 = \alpha\) for all \(\alpha \in R[i]\text{.}\)
  4. Show that \(R[i]\) has inverses: For each element \(\alpha \in R[i]\text{,}\) there exists an element \(-\alpha \in R[i]\) such that \(\alpha + (-\alpha) = 0\text{.}\)
  5. Show that multiplication is associative: \((\alpha\cdot\beta) \cdot \gamma = \alpha\cdot(\beta\cdot\gamma)\text{.}\)
  6. Show that there is an element \(1 \in R[i]\) such that \(1 \cdot \alpha = \alpha \cdot 1 = \alpha\text{.}\)
  7. Show that \(\alpha\cdot(\beta+\gamma)=\alpha\cdot\beta + \alpha\cdot\gamma\) for all \(\alpha,\beta,\gamma \in R[i]\text{.}\)

9.

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

10.

In this exercise, \(F\) will be a field with no square root of \(-1\text{,}\) i.e. such that there is no element \(r \in F\) such that \(r^2 + 1 = 0\text{.}\)
  1. Show that if \(a, b \in F\) such that \(a^2 + b^2 =0\) then \(a = b = 0\text{.}\)
  2. Show that the ring \(F[i]\) is a field.
  3. Using Exercise 3.4.10, conclude that \(\mathbb Z_p[i]\) is a field with \(p^2\) elements whenever \(p\) is a prime with \(p \equiv 3 \pmod{4}\text{.}\)

11.

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.

12.

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

13.

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 \(R = \{ 0 \}\text{.}\)

14.

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 \(r, s \in S\text{.}\)

15.

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.

16.

Prove that the only subring of \(\mathbb Z\) is \(\mathbb Z\text{.}\)

17.

Let \(R\) be a ring of characteristic \(n \gt 0\text{.}\) For any \(r \in R\text{,}\) show that \(r + \cdots + r\) (n times) is equal to 0.

18.

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 \(R\text{.}\)

19.

Prove that the center of the ring \(M_2(\mathbb R)\) is the set of matrices \(\{aI_2 : a \in \mathbb R\}\text{,}\) where \(I_2\) is the \(2\times 2\) identity matrix.

20.

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

21.

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

22.

Prove that \(\mathbb Q[\sqrt{2}]\) from Exercise 4.5.1 is not isomorphic to \(\mathbb Q\)

23.

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

24.

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.

25.

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 \(p\text{.}\)

26.

The following problems concern the ring of quaternions \(\mathbb H\) as in Example 4.7.
  1. Compute the products \((1+i)(1+i+j)\) and \((1+i+j)(1+i)\text{.}\)
  2. Find the inverse of \(1+k\text{.}\)

27.

Let \(\mathbb H\) be the ring of quaternions from Example 4.7. Let \(x, y, z\) be real numbers such that \(x^2+y^2+z^2=1\) and set \(\alpha = x\mathbf i + y \mathbf j + z \mathbf k\text{.}\)
  1. Show that \(\alpha^2 = -1\text{.}\)
  2. For \(a\in\mathbb R\text{,}\) show that \(\alpha a = a \alpha\text{.}\)
  3. Define \(\phi\colon\mathbb C \rightarrow \mathbb H\) by \(\phi(a+bi)=a+b\alpha\text{,}\) where \(a,b \in \mathbb R\) and show that \(\phi\) is a injective ring homomorphism.

28.

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

29.

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

30.

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.

31.

If \(x\) is an idempotent in a ring \(R\) and \(\phi\colon R \rightarrow S\) is a ring homomorphism, then show that \(\phi(x)\) is an idempotent in \(S\text{.}\)

32.

Suppose that \(x\) and \(y\) are idempotents in a commutative ring R.
  1. Show that \(1-x\) is an idempotent.
  2. Show that \(xy\) is an idempotent.
  3. Show that \(x + y -xy\) is an idempotent.

33.

Find all idempotents in the ring \(\mathbb Z_{12}\text{.}\)

34.

If \(m\) and \(k\) are relatively prime integers greater than 1 and \(n=mk\text{,}\) then show that \(\mathbb Z_n\) has at least four idempotents (including 0 and 1).

35.

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 \({\mathbb Z}_p\text{.}\)

36.

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 \(F_D\text{.}\)

37.

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 \(S\text{.}\)
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