§3.2 Exercises

Exercises 3.2.A

Exercise 3.2.1

Let \( R \) be a ring and \( a, b \in R \).

  • (a) \((a + b)(a - b) = ?\)
  • (b) \((a + b)^3 = ?\)
  • (c) What are the answers in parts (a) and (b) if \( R \) is commutative?

Exercise 3.2.2

Find the inverse of matrices \( A \), \( B \), and \( C \) in Example 3.2.7.

Exercise 3.2.3

An element \( e \) of a ring \( R \) is said to be idempotent if \( e^2 = e \).

  • (a) Find four idempotent elements in the ring \( M(\mathbb{R}) \).
  • (b) Find all idempotents in \( \mathbb{Z}_{12} \).

Exercise 3.2.4

For each matrix \( A \), find a matrix \( C \) such that \( AC = 0 \) or \( CA = 0 \):

\[ A = \begin{pmatrix} 6 & 9 \\ 2 & 3 \end{pmatrix}, \quad A = \begin{pmatrix} -5 & -2 \\ -10 & 4 \end{pmatrix}, \quad A = \begin{pmatrix} \frac{1}{2} & \frac{1}{4} \\ \frac{3}{2} & \frac{3}{4} \end{pmatrix} \]

Exercise 3.2.5

  • (a) Show that a ring has only one zero element. [Hint: If there were more than one, how many solutions would the equation \( 0_R + x = 0_R \) have?]
  • (b) Show that a ring \( R \) with identity has only one identity element.
  • (c) Can a unit in a ring \( R \) with identity have more than one inverse? Why?

Exercise 3.2.6

  • (a) Suppose \( A \) and \( C \) are nonzero matrices in \( M(\mathbb{R}) \) such that \( AC = 0 \). If \( k \) is any real number, show that \( A(kC) = 0 \), where \( kC \) is the matrix \( C \) with every entry multiplied by \( k \). Hence, the equation \( AX = 0 \) has infinitely many solutions.
  • (b) If \( A = \begin{pmatrix} 1 & 3 \\ 2 & 6 \end{pmatrix} \), find four solutions of the equation \( AX = 0 \).

Exercise 3.2.7

Let \(R\) be a ring with identity and let \(S = \{nr \mid n \in \mathbb{Z}\}\). Prove that \(S\) is a subring of \(R\).

[The definition of \(na\) with \(n \in \mathbb{Z}, a \in R\) is on page 62. Also see Exercise 3.2.27.]

Exercise 3.2.8

Let \(R\) be a ring and \(b\) a fixed element of \(R\). Let \(T = \{rb \mid r \in R\}\). Prove that \(T\) is a subring of \(R\).

Exercise 3.2.9

Show that the set \(S\) of matrices of the form \(\begin{pmatrix} b & 4b \\ b & a \end{pmatrix}\) with \(a\) and \(b\) real numbers is a subring of \(M(\mathbb{R})\).

Exercise 3.2.10

Let \(R\) and \(S\) be rings and consider these subsets of \(R \times S\):

\[ \overline{R} = \{(r, 0_S) \mid r \in R\}, \quad \overline{S} = \{(0_R, s) \mid s \in S\}. \]
  • (a) If \(R = \mathbb{Z}_3\) and \(S = \mathbb{Z}_5\), what are the sets \(\overline{R}\) and \(\overline{S}\)?
  • (b) For any rings \(R\) and \(S\), show that \(\overline{R}\) is a subring of \(R \times S\).
  • (c) For any rings \(R\) and \(S\), show that \(\overline{S}\) is a subring of \(R \times S\).

Exercise 3.2.11

Let \(R\) be a ring and \(m\) a fixed integer. Let \(S = \{r \in R \mid mr = 0_R\}\). Prove that \(S\) is a subring of \(R\).

Exercise 3.2.12

  • (a) Prove that the equation \(a + x = b\) has a unique solution in \(R\). [You must prove that there is a solution and that this solution is the only one.]
  • (b) If \(R\) is a ring with identity and \(a\) is a unit, prove that the equation \(ax = b\) has a unique solution in \(R\).

Exercise 3.2.13

Let \(S\) and \(T\) be subrings of a ring \(R\). In (a) and (b), if the answer is "yes," prove it. If the answer is "no," give a counterexample.

  • (a) Is \(S \cap T\) a subring of \(R\)?
  • (b) Is \(S \cup T\) a subring of \(R\)?

Exercise 3.2.14

Prove that the only idempotents in an integral domain \(R\) are \(0_R\) and \(1_R\).

[See Exercise 3.2.3.]

Exercise 3.2.15

  • (a) If \(a\) and \(b\) are units in a ring \(R\) with identity, prove that \(ab\) is a unit whose inverse is \((ab)^{-1} = b^{-1}a^{-1}\).
  • (b) Give an example to show that if \(a\) and \(b\) are units, then \(a^{-1}b^{-1}\) need not be the multiplicative inverse of \(ab\).

Exercise 3.2.16

Prove or disprove: The set of units in a ring \(R\) with identity is a subring of \(R\).

Exercise 3.2.17

If \(u\) is a unit in a ring \(R\) with identity, prove that \(u\) is not a zero divisor.

Exercise 3.2.18

Let \(a\) be a nonzero element of a ring \(R\) with identity. If the equation \(ax = 1_R\) has a solution \(u\) and the equation \(ya = 1_R\) has a solution \(v\), prove that \(u = v\).

Exercise 3.2.19

Let \(R\) and \(S\) be rings with identity. What are the units in the ring \(R \times S\)?

Exercise 3.2.20

Let \(R\) and \(S\) be nonzero rings (meaning that each of them contains at least one nonzero element). Show that \(R \times S\) contains zero divisors.

Exercise 3.2.21

Let \(R\) be a ring and let \(a\) be a nonzero element of \(R\) that is not a zero divisor. Prove that cancelation holds for \(a\); that is, prove that

  • (a) If \(ab = ac\) in \(R\), then \(b = c\).
  • (b) If \(ba = ca\) in \(R\), then \(b = c\).

Exercise 3.2.22

  • (a) If \(ab\) is a zero divisor in a ring \(R\), prove that \(a\) or \(b\) is a zero divisor.
  • (b) If \(a\) or \(b\) is a zero divisor in a commutative ring \(R\) and \(ab \neq 0_R\), prove that \(ab\) is a zero divisor.

Exercise 3.2.23

(a) Let \(R\) be a ring and \(a, b \in R\). Let \(m\) and \(n\) be nonnegative integers and prove that:

  • (i) \((m + n)a = ma + na\).
  • (ii) \(m(a + b) = ma + mb\).
  • (iii) \(m(ab) = (ma)b = a(mb)\).
  • (iv) \((ma)(nb) = mn(ab)\).

(b) Do part (a) when \(m\) and \(n\) are any integers.

Exercise 3.2.24

Let \(R\) be a ring and \(a, b \in R\). Let \(m\) and \(n\) be positive integers.

  • (a) Show that \(a^m a^n = a^{m+n}\) and \((a^m)^n = a^{mn}\).
  • (b) Under what conditions is it true that \((ab)^n = a^n b^n\)?

Exercise 3.2.25

Let \(S\) be a subring of a ring \(R\) with identity.

  • (a) If \(S\) has an identity, show by example that \(1_S\) may not be the same as \(1_R\).
  • (b) If both \(R\) and \(S\) are integral domains, prove that \(1_S = 1_R\).

Exercises 3.2.B

Exercise 3.2.26

Let \(S\) be a subring of a ring \(R\). Prove that \(0_S = 0_R\).

Hint: For \(a \in S\), consider the equation \(a + x = a\).

Exercise 3.2.27

Let \(R\) be a ring with identity and \(b\) a fixed element of \(R\) and let \(S = \{nb \mid n \in \mathbb{Z}\}\). Is \(S\) necessarily a subring of \(R\)?

[See Exercise 3.2.7 for the case when \(b = 1_R\).]

Exercise 3.2.28

Assume that \(R = \{0_R, 1_R, a, b\}\) is a ring and that \(a\) and \(b\) are units. Write out the multiplication table of \(R\).

Exercise 3.2.29

Let \(R\) be a commutative ring with identity. Prove that \(R\) is an integral domain if and only if cancelation holds in \(R\); that is, \(a \neq 0_R\) and \(ab = ac\) in \(R\) imply \(b = c\).

Exercise 3.2.30

Let \(R\) be a commutative ring with identity and \(b \in R\). Let \(T\) be the subring of all multiples of \(b\) (as in Exercise 3.2.8). If \(u\) is a unit in \(R\) and \(u \in T\), prove that \(T = R\).

Exercise 3.2.31

A Boolean ring is a ring \(R\) with identity in which \(x^2 = x\) for every \(x \in R\). For examples, see Exercises 3.1.19 and 3.1.44. If \(R\) is a Boolean ring, prove that:

  • (a) \(a + a = 0_R\) for every \(a \in R\), which means that \(a = -a\). Hint: Expand \((a + a)^2\).
  • (b) \(R\) is commutative. Hint: Expand \((a + b)^2\).

Exercise 3.2.32

Let \(R\) be a ring without identity. Let \(T\) be the set \(R \times \mathbb{Z}\). Define addition and multiplication in \(T\) by these rules:

\[ (r, m) + (s, n) = (r + s, m + n), \quad (r, m)(s, n) = (rs + ms + nr, mn). \]
  • (a) Prove that \(T\) is a ring with identity.
  • (b) Let \(\overline{R}\) consist of all elements of the form \((r, 0)\) in \(T\). Prove that \(\overline{R}\) is a subring of \(T\).

Exercise 3.2.33

Let \(R\) be a ring with identity. If \(ab = ad\) and \(a\) are units in \(R\), prove that \(b = d\).

Exercise 3.2.34

Let \(F\) be a field and \(A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\) a matrix in \(M(F)\).

  • (a) Prove that \(A\) is invertible if and only if \(ad - bc \neq 0_F\). Hint: See Examples 3.2.7, 3.2.8, and 3.2.10, and Exercise 3.2.17.
  • (b) Prove that \(A\) is a zero divisor if and only if \(ad - bc = 0_F\).

Exercise 3.2.35

Let \(A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\) be a matrix with integer entries.

  • (a) If \(ad - bc = \pm 1\), show that \(A\) is invertible in \(M(\mathbb{Z})\). Hint: See Example 3.1.7.
  • (b) If \(ad - bc \neq 0, 1, -1\), show that \(A\) is neither a unit nor a zero divisor in \(M(\mathbb{Z})\). Hint: Show that \(A\) has an inverse in \(M(\mathbb{R})\) that is not in \(M(\mathbb{Z})\). See Exercise 3.2.5(c). For zero divisors, see Exercise 3.2.34(b) and Example 3.1.10.

Exercise 3.2.36

Let \(R\) be a commutative ring with identity. Then the set \(M(R)\) of \(2 \times 2\) matrices with entries in \(R\) is a ring with identity by Exercise 3.1.37.

If \(A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in M(R)\) and \(ad - bc\) is a unit in \(R\), show that \(A\) is invertible in \(M(R)\). Hint: Replace \(\frac{1}{ad - bc}\) by \((ad - bc)^{-1}\) in Example 3.1.7.

Exercise 3.2.37

Let \(R\) be a ring with identity and \(a, b \in R\). Assume that \(a\) is not a zero divisor. Prove that \(ab = 1_R\) if and only if \(ba = 1_R\). Hint: Note that both \(ab = 1_R\) and \(ba = 1_R\) imply \(aba = a\) (why?); use Exercise 3.2.21.

Exercise 3.2.38

Let \(R\) be a ring with identity and \(a, b \in R\). Assume that neither \(a\) nor \(b\) is a zero divisor. If \(ab\) is a unit, prove that \(a\) and \(b\) are units. Hint: See Exercise 3.2.21.

Exercise 3.2.39

  • (a) If \(R\) is a finite commutative ring with identity and \(a \in R\), prove that \(a\) is either a zero divisor or a unit. Hint: If \(a\) is not a zero divisor, adapt the proof of Theorem 3.8, using Exercise 3.2.21.
  • (b) Is part (a) true if \(R\) is infinite? Justify your answer.

Exercise 3.2.40

An element \(a\) of a ring is nilpotent if \(a^n = 0_R\) for some positive integer \(n\). Prove that \(R\) has no nonzero nilpotent elements if and only if \(0_R\) is the unique solution of the equation \(x^2 = 0_R\).

Exercise 3.2.41

The following definition is needed for Exercises 3.2.41–3.2.43. Let \(R\) be a ring with identity. If there is a smallest positive integer \(n\) such that \(n \cdot 1_R = 0_R\), then \(R\) is said to have characteristic \(n\). If no such \(n\) exists, \(R\) is said to have characteristic zero.

  • (a) Show that \(\mathbb{Z}\) has characteristic zero and \(\mathbb{Z}_n\) has characteristic \(n\).
  • (b) What is the characteristic of \(\mathbb{Z}_4 \times \mathbb{Z}_7\)?

Exercise 3.2.42

Prove that a finite ring with identity has characteristic \(n\) for some \(n > 0\).

Exercise 3.2.43

Let \(R\) be a ring with identity of characteristic \(n > 0\).

  • (a) Prove that \(na = 0_R\) for every \(a \in R\).
  • (b) If \(R\) is an integral domain, prove that \(n\) is prime.

Exercises 3.2.C

Exercise 3.2.44

  • (a) Let \(a\) and \(b\) be nilpotent elements of a ring \(R\). Show that \(a + b\) is nilpotent.
  • (b) Let \(V\) be the set of all nilpotent elements of \(R\). Show that \(V\) is a subring of \(R\).

Exercise 3.2.45

Let \(R\) be a ring such that \(x^3 = x\) for every \(x \in R\). Prove that \(R\) is commutative.

Exercise 3.2.46

Let \(R\) be a nonzero finite commutative ring with no zero divisors. Prove that \(R\) is a field.