ยง3.3 Exercises

Exercises 3.3.A

Exercise 3.3.1

Let \( f: \mathbb{Z}_6 \to \mathbb{Z}_2 \times \mathbb{Z}_3 \) be the bijection given by

\[ 0 \to (0,0), \, 1 \to (1,1), \, 2 \to (0,2), \, 3 \to (1,0), \, 4 \to (0,1), \, 5 \to (1,2). \]

Use the addition and multiplication tables of \( \mathbb{Z}_6 \) and \( \mathbb{Z}_2 \times \mathbb{Z}_3 \) to show that \( f \) is an isomorphism.

Exercise 3.3.2

Use tables to show that \( \mathbb{Z}_2 \times \mathbb{Z}_2 \) is isomorphic to the ring \(R\) of the earlier Exercise 3.1.2.

Exercise 3.3.3

Let \( R \) be a ring and let \( R^* \) be the subring of \( R \times R \) consisting of all elements of the form \( (a, 0) \). Show that the function \( f:R \to R^* \) given by \( f(a) = (a,a) \) is an isomorphism.

Exercise 3.3.4

Let \( S \) be the subring \( \{0, 2, 4, 6, 8\} \) of \( \mathbb{Z}_{10} \) and let \( Z_5 = \{\overline{0}, \overline{1}, \overline{2}, \overline{3}, \overline{4}\} \) (notation as in Example 3.3.1). Show that the following bijection from \( Z_5 \) to \( S \) is not an isomorphism:

\[ \overline{0} \to 0, \, \overline{1} \to 2, \, \overline{2} \to 4, \, \overline{3} \to 6, \, \overline{4} \to 8. \]

Exercise 3.3.5

Prove that the field \( \mathbb{R} \) of real numbers is isomorphic to the ring of all \( 2 \times 2 \) matrices of the form

\[ \begin{pmatrix} 0 & a \\ 0 & a \end{pmatrix} , \]

with \( a \in \mathbb{R} \). [Hint: Consider the function \( f \) given by \( f(a) = \begin{pmatrix} 0 & a \\ 0 & a \end{pmatrix} \).]

Exercise 3.3.6

Let \( R \) and \( S \) be rings and let \( R \) be the subring of \( R \times S \) consisting of all elements of the form \( (a, 0_S) \). Show that the function \( f: R \to R \) given by \( f(a) = (a, 0_S) \) is an isomorphism.

Exercise 3.3.7

Prove that \( R \) is isomorphic to the ring \( S \) of all \( 2 \times 2 \) matrices of the form

\[ \begin{pmatrix} a & 0 \\ 0 & a \end{pmatrix} , \]

where \( a \in \mathbb{R} \).

Exercise 3.3.8

Let \( \mathbb{Q}(\sqrt{2}) \) be as in the earlier Exercise 3.1.39. Prove that the function \( f: \mathbb{Q}(\sqrt{2}) \to \mathbb{Q}(\sqrt{2}) \) given by \( f(a + b\sqrt{2}) = a - b\sqrt{2} \) is an isomorphism.

Exercise 3.3.9

If \( f: \mathbb{Z} \to \mathbb{Z} \) is an isomorphism, prove that \( f \) is the identity map. [Hint: What are \( f(1), f(1 + 1), \ldots \)?]

Exercise 3.3.10

If \(R\) is a ring with identity and \(f: R \to S\) is a homomorphism from \(R\) to a ring \(S\), prove that \(f(1_R)\) is an idempotent in \(S\). [Idempotents were defined in the earlier Exercise 3.2.3.]

Exercise 3.3.11

State at least one reason why the given function is not a homomorphism.

  • (a) \(f: \mathbb{R} \to \mathbb{R}\) and \(f(x) = \sqrt{x}\).
  • (b) \(g: E \to E\), where \(E\) is the ring of even integers and \(f(x) = 3x\).
  • (c) \(h: \mathbb{R} \to \mathbb{R}\) and \(f(x) = 2^x\).
  • (d) \(k: \mathbb{Q} \to \mathbb{Q}\), where \(k(0) = 0\) and \(k\left(\frac{a}{b}\right) = \frac{b}{a}\) if \(a \neq 0\).

Exercise 3.3.12

Which of the following functions are homomorphisms?

  • (a) \(f: \mathbb{Z} \to \mathbb{Z}\), defined by \(f(x) = -x\).
  • (b) \(f: \mathbb{Z}_2 \to \mathbb{Z}_2\), defined by \(f(x) = -x\).
  • (c) \(g: \mathbb{Q} \to \mathbb{Q}\), defined by \(g(x) = \frac{1}{x^2 + 1}\).
  • (d) \(h: \mathbb{R} \to M(\mathbb{R})\), defined by \(h(a) = \begin{pmatrix} -a & 0 \\ a & 0 \end{pmatrix}\).
  • (e) \(f: \mathbb{Z}_{12} \to \mathbb{Z}_4\), defined by \(f([x]_{12}) = [x]_4\), where \([u]_n\) denotes the class of the integer \(u\) in \( \mathbb{Z}_n \).

Exercise 3.3.13

Let \( R \) and \( S \) be rings.

  • (a) Prove that \( f: R \times S \to R \) given by \( f((r, s)) = r \) is a surjective homomorphism.
  • (b) Prove that \( g: R \times S \to S \) given by \( g((r, s)) = s \) is a surjective homomorphism.
  • (c) If both \( R \) and \( S \) are nonzero rings, prove that the homomorphisms \(f\) and \(g\) are not injective.

Exercise 3.3.14

Let \( f: \mathbb{Z} \to \mathbb{Z}_6 \) be the homomorphism in Example 3.3.6. Let \( K = \{a \in \mathbb{Z} \mid f(a) = [0]\} \). Prove that \( K \) is a subring of \( \mathbb{Z} \).

Exercise 3.3.15

Let \( f: R \to S \) be a homomorphism of rings. If \( r \) is a zero divisor in \( R \), is \( f(r) \) a zero divisor in \( S \)?

Exercises 3.3.B

Exercise 3.3.16

Let \( T \), \( R \), and \( F \) be the four-element rings whose tables are given in Example 3.1.5 and in Exercises 3.1.2 and 3.1.3. Show that no two of these rings are isomorphic.

Exercise 3.3.17

Show that the complex conjugation function \( f: \mathbb{C} \to \mathbb{C} \) (whose rule is \( f(a + bi) = a - bi \) is a bijection.

Exercise 3.3.18

Show that the isomorphism of \( \mathbb{Z}_5 \) and \( S \) in Example 3.3.1 is given by the function whose rule is \( f([x]_5) = [6x]_{10} \) (notation as in Exercise 3.3.12(e)). Give a direct proof (without using tables) that this map is a homomorphism.

Exercise 3.3.19

Show that \(S = \{0, 4, 8, 12, 16, 20, 24\}\) is a subring of \(\mathbb{Z}_{28}\). Then prove that the map \(f: \mathbb{Z}_7 \to S\) given by \(f([x]_7) = [8x]_{28}\) is an isomorphism.

Exercise 3.3.20

Let \(E\) be the ring of even integers with the \(*\) multiplication defined in the earlier Exercise 3.1.23. Show that the map \(f: E \to \mathbb{Z}\) given by \(f(x) = x/2\) is an isomorphism.

Exercise 3.3.21

Let \(\mathbb{Z}^*\) denote the ring of integers with the \(\oplus\) and \(\odot\) operations defined in the earlier Exercise 3.1.22. Prove that \(\mathbb{Z}\) is isomorphic to \(\mathbb{Z}^*\).

Exercise 3.3.22

Let \(\mathbb{Z}\) denote the ring of integers with the \(\oplus\) and \(\odot\) operations defined in Exercise 3.1.24. Prove that \(\mathbb{Z}\) is isomorphic to \(\mathbb{Z}\).

Exercise 3.3.23

Let \(C\) be the field of the earlier Exercise 3.1.45. Show that \(C\) is isomorphic to the field \(\mathbb{C}\) of complex numbers.

Exercise 3.3.24

(a) Let \(A\) be the set \(R \times R\) with the usual coordinate-wise addition, as in Theorem 3.1. Define a new multiplication by the rule \((a, b)(c, d) = (ac, bd)\). Show that \(A\) is a ring.

(b) Show that the ring of part (a) is isomorphic to the ring of all matrices in \(M(R)\) of the form \(\begin{pmatrix} a & 0 \\ 0 & b \end{pmatrix}\).

Exercise 3.3.25

Let \(Z\) be the ring of all matrices in \(M(Z)\) of the form \(\begin{pmatrix} a & 0 \\ b & c \end{pmatrix}\). Show that the function \(f: Z \to Z\) given by \(f\left(\begin{pmatrix} a & 0 \\ b & c \end{pmatrix}\right) = a\) is a surjective homomorphism but not an isomorphism.

Exercise 3.3.26

Show that the homomorphism \(g\) in Example 3.3.7 is injective but not surjective.

Exercise 3.3.27

(a) If \(g: R \to S\) and \(f: S \to T\) are homomorphisms, show that \(f \circ g: R \to T\) is a homomorphism.

(b) If \(f\) and \(g\) are isomorphisms, show that \(f \circ g\) is also an isomorphism.

Exercise 3.3.28

(a) Give an example of a homomorphism \(f: R \to S\) such that \(S\) has an identity but \(R\) does not.

(b) Give an example of a homomorphism \(f: R \to S\) such that \(S\) has an identity and \(R\) does not. Does this contradict part (4) of Theorem 3.10?

Exercise 3.3.29

Let \(f: R \to S\) be an isomorphism of rings and let \(g: S \to R\) be the inverse function of \(f\) (as defined in Appendix B). Show that \(g\) is also an isomorphism.

Hint: To show \(g(a + b) = g(a) + g(b)\), consider the images of the left- and right-hand sides under \(f\) and use the facts that \(f\) is a homomorphism and \(f \circ g\) is the identity map.

Exercise 3.3.30

Let \(f: R \to S\) be a homomorphism of rings and let \(K = \{r \in R \mid f(r) = 0_S\}\). Prove that \(K\) is a subring of \(R\).

Exercise 3.3.31

Let \(f: R \to S\) be a homomorphism of rings and \(T\) a subring of \(S\). Let \(P = \{r \in R \mid f(r) \in T\}\). Prove that \(P\) is a subring of \(R\).

Exercise 3.3.32

Assume \(n \equiv 1 \mod m\). Show that the function \(f: \mathbb{Z}_m \to \mathbb{Z}_m\) given by \(f([x]_m) = [nx]_m\) is an injective homomorphism but not an isomorphism when \(n \geq 2\) (notation as in Exercise 3.3.12(e)).

Exercise 3.3.33

(a) Let \(\theta\) be the ring of functions from \(R\) to \(R\), as in Example 3.1.8. Let \(\theta: T \to R\) be the function defined by \(\theta(f) = f(5)\). Prove that \(\theta\) is a surjective homomorphism. Is \(\theta\) an isomorphism?

(b) Is part (a) true if 5 is replaced by any constant \(c \in R\)?

Exercise 3.3.34

If \(f: R \to S\) is an isomorphism of rings, which of the following properties are preserved by this isomorphism? Justify your answers.

  • (a) \(a \in R\) is a zero divisor.

Exercise 3.3.35

Show that the first ring is not isomorphic to the second.

  • (a) \(E\) and \(\mathbb{Z}\)
  • (b) \(R \times R \times R \times R\) and \(M(R)\)
  • (c) \(\mathbb{Z}_4 \times \mathbb{Z}_4\) and \(\mathbb{Z}_{16}\)
  • (d) \(Q\) and \(R\)
  • (e) \(\mathbb{Z} \times \mathbb{Z}_4\) and \(\mathbb{Z}\)
  • (f) \(\mathbb{Z}_4 \times \mathbb{Z}_4\) and \(\mathbb{Z}_{16}\)

Exercise 3.3.36

(a) If \(f: R \to S\) is a homomorphism of rings, show that for any \(r \in R\) and \(n \in \mathbb{Z}, f(nr) = nf(r)\).

(b) Prove that isomorphic rings with identity have the same characteristic. [See Exercises 3.2.41-3.2.43.]

(c) If \(f: R \to S\) is a homomorphism of rings with identity, is it true that \(R\) and \(S\) have the same characteristic?

Exercise 3.3.37

  • (a) Assume that \(e\) is a nonzero idempotent in a ring \(R\) and that \(e\) is not a zero divisor. Prove that \(e\) is the identity element of \(R\). Hint: \(e^2 = e\) (Why?). If \(a \in R\), multiply both sides of \(e^2 = e\) by \(a\).
  • (b) Let \(S\) be a ring with identity and \(T\) a ring with no zero divisors. Assume that \(f: S \to T\) is a nonzero homomorphism of rings (meaning that at least one element of \(S\) is not mapped to \(0_T\)). Prove that \(f(1_S)\) is the identity element of \(T\). Hint: Show that \(f(1_S)\) satisfies the hypotheses of part (a).

Exercise 3.3.38

Let \(F\) be a field and \(f: F \to R\) a homomorphism of rings.

  • (a) If there is a nonzero element \(c\) of \(F\) such that \(f(c) = 0_R\), show that \(f\) is the zero homomorphism (that is, \(f(x) = 0_R\) for every \(x \in F\)). Hint: \(c^{-1}\) exists (Why?). If \(x \in F\), consider \(f(xc^{-1})\).
  • (b) Prove that \(f\) is either injective or the zero homomorphism. Hint: If \(f\) is not the zero homomorphism and \(f(a) = f(b)\), then \(f(a - b) = 0_R\).

Exercise 3.3.39

Let \(R\) be a ring without identity. Let \(T\) be the ring with identity of the earlier Exercise 3.2.32. Show that \(R\) is isomorphic to the subring of \(T\). Thus, if \(R\) is identified with \(R\), then \(R\) is a subring of a ring with identity.

Exercises 3.3.C

Exercise 3.3.40

For each positive integer \(k\), let \(k\mathbb{Z}\) denote the ring of all integer multiples of \(k\) (see the earlier Exercise 3.1.6). Prove that if \(m \neq n\), then \(m\mathbb{Z}\) is not isomorphic to \(n\mathbb{Z}\).

Exercise 3.3.41

Let \(m, n \in \mathbb{Z}\) with \((m, n) = 1\) and let \(f: \mathbb{Z}_m \times \mathbb{Z}_n \to \mathbb{Z}_{mn}\) be the function given by \(f([a]_m, [a]_n) = [a]_{mn}\). (Notation as in Exercise 3.3.12(e). Example 3.3.8 is the case \(m = 3, n = 4\).)

  • (a) Show that the map \(f\) is well defined, that is, show that if \([a]_m = [b]_m\) in \(\mathbb{Z}_m\) and \([a]_n = [b]_n\) in \(\mathbb{Z}_n\), then \([a]_{mn} = [b]_{mn}\) in \(\mathbb{Z}_{mn}\).
  • (b) Prove that \(f\) is an isomorphism. Hint: Adapt the proof in Example 3.3.8; the difference is that proving \(f\) is a bijection takes more work here.

Exercise 3.3.42

If \((m, n) \neq 1\), prove that \(\mathbb{Z}_{mn}\) is not isomorphic to \(\mathbb{Z}_m \times \mathbb{Z}_n\).