§3.1 Exercises

Exercise 3.1.1

The following subsets of \(\mathbb{Z}\) (with ordinary addition and multiplication) satisfy all but one of the axioms for a ring. In each case, which axiom fails?

• (a) The set \(S\) of all odd integers and 0.
• (b) The set of nonnegative integers.

Exercise 3.1.2

Let \(R = \{0, e, b, c\}\) with addition and multiplication defined by the tables on page 54. Assume associativity and distributivity and show that \(R\) is a ring with identity. Is \(R\) commutative? Is \(R\) a field?

Exercise 3.1.3

Let \(F = \{0, e, a, b, c\}\) with operations given by the following tables. Assume associativity and distributivity and show that \(F\) is a field.

Exercise 3.1.4

Find matrices \(A\) and \(C\) in \(M(\mathbb{R})\) such that \(AC = 0\), but \(CA \neq 0\), where 0 is the zero matrix. Hint: Example 3.1.6.

Exercise 3.1.5

Which of the following six sets are subrings of \(M(\mathbb{R})\)? Which ones have an identity?

• (a) All matrices of the form \(\begin{pmatrix} r & 0 \\ 0 & 0 \end{pmatrix}\) with \(r \in \mathbb{Q}\).
• (b) All matrices of the form \(\begin{pmatrix} a & b \\ c & 0 \end{pmatrix}\) with \(a, b, c \in \mathbb{Z}\).
• (c) All matrices of the form \(\begin{pmatrix} a & b \\ c & 0 \end{pmatrix}\) with \(a, b, c \in \mathbb{R}\).
• (d) All matrices of the form \(\begin{pmatrix} a & 0 \\ a & 0 \end{pmatrix}\) with \(a \in \mathbb{R}\).
• (e) All matrices of the form \(\begin{pmatrix} a & 0 \\ 0 & a \end{pmatrix}\) with \(a \in \mathbb{R}\).
• (f) All matrices of the form \(\begin{pmatrix} a & 0 \\ 0 & 0 \end{pmatrix}\) with \(a \in \mathbb{R}\).

Exercise 3.1.6

• (a) Show that the set \(R\) of all multiples of 3 is a subring of \(\mathbb{Z}\).
• (b) Let \(k\) be a fixed integer. Show that the set of all multiples of \(k\) is a subring of \(\mathbb{Z}\).

Exercise 3.1.7

Let \(K\) be the set of all integer multiples of \(\sqrt{2}\), that is, all real numbers of the form \(n\sqrt{2}\) with \(n \in \mathbb{Z}\). Show that \(K\) satisfies Axioms 1–5, but is not a ring.

Exercise 3.1.8

Is the subset \(\{1, -1, i, -i\}\) a subring of \(\mathbb{C}\)?

Exercise 3.1.9

Let \(R\) be a ring and consider the subset \(R^*\) of \(R \times R\) defined by \(R^* = \{(r, r) \mid r \in R\}\).

• (a) If \(R = \mathbb{Z}_6\), list the elements of \(R^*\).
• (b) For any ring \(R\), show that \(R^*\) is a subring of \(R \times R\).

Exercise 3.1.10

Is \(S = \{(a, b) \mid a + b = 0\}\) a subring of \(\mathbb{Z} \times \mathbb{Z}\)? Justify your answer.

Exercise 3.1.11

Let \(S\) be the subset of \(M(\mathbb{R})\) consisting of all matrices of the form \(\begin{pmatrix} a & a \\ b & b \end{pmatrix}\).

• (a) Prove that \(S\) is a ring.
• (b) Show that \(J = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\) is a right identity in \(S\) (meaning that \(AJ = A\) for every \(A\) in \(S\)).
• (c) Show that \(J\) is not a left identity in \(S\) by finding a matrix \(B\) in \(S\) such that \(JB \neq B\).

For more information about \(S\), see Exercise 41.

Exercise 3.1.12

Let \(\mathbb{Z}[i]\) denote the set \(\{a + bi \mid a, b \in \mathbb{Z}\}\). Show that \(\mathbb{Z}[i]\) is a subring of \(\mathbb{C}\).

Exercise 3.1.13

Let \(\mathbb{Z}[\sqrt{2}]\) denote the set \(\{a + b\sqrt{2} \mid a, b \in \mathbb{Z}\}\). Show that \(\mathbb{Z}[\sqrt{2}]\) is a subring of \(\mathbb{R}\). [See Example 3.1.20.]

Exercise 3.1.14

Let \(T\) be the ring in Example 3.1.8. Let \(S = \{f \in T \mid f(2) = 0\}\). Prove that \(S\) is a subring of \(T\).

Exercise 3.1.15

Write out the addition and multiplication tables for:

• (a) \(\mathbb{Z}_2 \times \mathbb{Z}_3\)
• (b) \(\mathbb{Z}_2 \times \mathbb{Z}_2\)
• (c) \(\mathbb{Z}_3 \times \mathbb{Z}_3\)

Exercise 3.1.16

Let \(A = \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}\) and \(0 = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}\) in \(M(\mathbb{R})\). Let \(S\) be the set of all matrices \(B\) such that \(AB = 0\).

• (a) List three matrices in \(S\). [Many correct answers are possible.]
• (b) Prove that \(S\) is a subring of \(M(\mathbb{R})\). [Hint: If \(B\) and \(C\) are in \(S\), show that \(B + C\) and \(BC\) are in \(S\) by computing \(A(B + C)\) and \(A(BC)\).]

Exercise 3.1.17

Define a new multiplication in \(\mathbb{Z}\) by the rule: \(ab = 0\) for all \(a, b \in \mathbb{Z}\). Show that with ordinary addition and this new multiplication, \(\mathbb{Z}\) is a commutative ring.

Exercise 3.1.18

Define a new multiplication in \(\mathbb{Z}\) by the rule: \(ab = 1\) for all \(a, b \in \mathbb{Z}\). With ordinary addition and this new multiplication, is \(\mathbb{Z}\) a ring?

Exercise 3.1.19

Let \(S = \{a, b, c\}\) and let \(P(S)\) be the set of all subsets of \(S\); denote the elements of \(P(S)\) as follows:

\(S = \{a, b, c\}; \quad D = \{a, b\}; \quad E = \{a, c\}; \quad F = \{b, c\}; A = \{a\}; \quad B = \{b\}; \quad C = \{c\}; \quad 0 = \varnothing.\)

Define addition and multiplication in \(P(S)\) by these rules:

\(M + N = (M - N) \cup (N - M) \quad and \quad (MN = M \cap N)\).

Write out the addition and multiplication tables for \(P(S)\). Also, see Exercise 44.

Exercise 3.1.20

Show that the subset \(R = \{0, 3, 6, 9, 12, 15\}\) of \(\mathbb{Z}_{18}\) is a subring. Does \(R\) have an identity?

Exercise 3.1.21

Show that the subset \(S = \{0, 2, 4, 6, 8\}\) of \(\mathbb{Z}_{10}\) is a subring. Does \(S\) have an identity?

Exercise 3.1.22

Define a new addition \(\oplus\) and multiplication \(\odot\) on \(\mathbb{Z}\) by:

\(a \oplus b = a + b - 1 \quad \text{and} \quad a \odot b = ab + a + b,\)

where the operations on the right-hand side of the equal signs are ordinary addition, subtraction, and multiplication. Prove that, with the new operations \(\oplus\) and \(\odot\), \(\mathbb{Z}\) is an integral domain.

Exercise 3.1.23

Let \(E\) be the set of even integers with ordinary addition. Define a new multiplication \(\star\) on \(E\) by the rule " \(a \star b = ab/2\) " (where the product on the right is ordinary multiplication). Prove that with these operations \(E\) is a commutative ring with identity.

Exercise 3.1.24

Define a new addition and multiplication on \(\mathbb{Z}\) by:

\(a \oplus b = a + b - 1 \quad \text{and} \quad a \odot b = ab - (a + b) + 2.\)

Prove that with these new operations \(\mathbb{Z}\) is an integral domain.

Exercise 3.1.25

Define a new addition and multiplication on \(\mathbb{Q}\) by:

\(r \oplus s = r + s + 1 \quad \text{and} \quad r \odot s = rs + r + s.\)

Prove that with these new operations \(\mathbb{Q}\) is a commutative ring with identity. Is it an integral domain?

Exercise 3.1.26

Let \(L\) be the set of positive real numbers. Define a new addition and multiplication on \(L\) by:

\(a \oplus b = ab \quad \text{and} \quad a \odot b = a^{\log b}.\)

• (a) Is \(L\) a ring under these operations? If so, does \(L\) have an identity?
• (b) Is \(L\) a commutative ring?
• (c) Is \(L\) a field?

Exercise 3.1.27

Let \(S\) be the set of rational numbers that can be written with an odd denominator. Prove that \(S\) is a subring of \(\mathbb{Q}\) but is not a field.

Exercise 3.1.28

Let \(p\) be a positive prime and let \(R\) be the set of all rational numbers that can be written in the form \(r/p^i\) with \(r, i \in \mathbb{Z}\), and \(i \geq 0\). Note that \(\mathbb{Z} \subseteq R\) because each \(n \in \mathbb{Z}\) can be written as \(n/p^0\). Show that \(R\) is a subring of \(\mathbb{Q}\).

Exercise 3.1.29

The addition table and part of the multiplication table for a three-element ring are given below. Use the distributive laws to complete the multiplication table.

Exercise 3.1.30

The addition table and part of the multiplication table for a four-element ring are given below. Use the distributive laws to complete the multiplication table.

Exercise 3.1.31

A **scalar matrix** in \(M(\mathbb{R})\) is a matrix of the form \(\begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix}\) for some real number \(k\).

• (a) Prove that the set of scalar matrices is a subring of \(M(\mathbb{R})\).
• (b) If \(K\) is a scalar matrix, show that \(KA = AK\) for every \(A\) in \(M(\mathbb{R})\).
• (c) If \(K\) is a matrix in \(M(\mathbb{R})\) such that \(KA = AK\) for every \(A\) in \(M(\mathbb{R})\), show that \(K\) is a scalar matrix. [Hint: If \(K = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\), let \(A = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}\). Use the fact that \(KA = AK\) to show that \(b = 0\) and \(c = 0\). Then make a similar argument with \(A = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}\) to show that \(a = d\).]

Exercise 3.1.32

Let \(R\) be a ring and let \(Z(R) = \{a \in R \mid ar = ra \text{ for every } r \in R\}\). In other words, \(Z(R)\) consists of all elements of \(R\) that commute with every other element of \(R\). Prove that \(Z(R)\) is a subring of \(R\). \(Z(R)\) is called the center of the ring \(R\). [Exercise 31 shows that the center of \(M(\mathbb{R})\) is the subring of scalar matrices.]

Exercise 3.1.33

Prove Theorem 3.1.

Exercise 3.1.34

Show that \(M(\mathbb{Z}_2\) (all \(2 \times 2\) matrices with entries in \(\mathbb{Z}_2\)) is a 16-element noncommutative ring with identity.

Exercise 3.1.35

Prove or disprove:

• (a) If \(R\) and \(S\) are integral domains, then \(R \times S\) is an integral domain.
• (b) If \(R\) and \(S\) are fields, then \(R \times S\) is a field.

Exercise 3.1.36

Let \( T \) be the ring in Example 3.1.8 and let \( f, g \) be given by:

Show that \( f, g \in T \) and that \( fg = 0_T \). Therefore, \( T \) is not an integral domain.

Exercise 3.1.37

(a) If \(R\) is a ring, show that the ring \(M(R)\) of all \(2 \times 2\) matrices with entries in \(R\) is a ring.

(b) If \(R\) has an identity, show that \(M(R)\) also has an identity.

Exercise 3.1.38

If \(R\) is a ring and \(a \in R\), let \(A_R = \{r \in R \mid ar = 0_R\}\). Prove that \(A_R\) is a subring of \(R\). \(A_R\) is called the right annihilator of \(a\). [For an example, see Exercise 3.1.16 in which the ring \(S\) is the right annihilator of the matrix \(A\).]

Exercise 3.1.39

Let \(\mathbb{Q}(\sqrt{2}) = \{r + s\sqrt{2} \mid r, s \in \mathbb{Q}\}\). Show that \(\mathbb{Q}(\sqrt{2})\) is a subfield of \(\mathbb{R}\).

[Hint: To show that the solution of \((r + s\sqrt{2})(r - s\sqrt{2}) = 1\) is actually in \(\mathbb{Q}(\sqrt{2})\), multiply \((r + s\sqrt{2})\) by \((r - s\sqrt{2})/(r - s\sqrt{2})\).]

Exercise 3.1.40

Let \(d\) be an integer that is not a perfect square. Show that \(\mathbb{Q}(\sqrt{d}) = \{a + b\sqrt{d} \mid a, b \in \mathbb{Q}\}\) is a subfield of \(\mathbb{C}\). [Hint: See Exercise 3.1.39.]

Exercise 3.1.41

Let \( S \) be the ring in Exercise 3.1.11.

(a) Verify that each of these matrices is a right identity in \( S \):

(b) Prove that the matrix \(\begin{pmatrix} x & y \\ y & x \end{pmatrix}\) is a right identity in \( S \) if and only if \( x + y = 1 \).

(c) If \( x + y = 1 \), show that \(\begin{pmatrix} x & y \\ y & x \end{pmatrix}\) is not a left identity in \( S \).

Exercise 3.1.42

A division ring is a (not necessarily commutative) ring \(R\) with identity \(1_R \neq 0_R\) that satisfies Axioms 11 and 12 (pages 48 and 49). Thus, a field is a commutative division ring. See Exercise 3.1.43 for a noncommutative example. Suppose \(R\) is a division ring and \(a, b\) are nonzero elements of \(R\).

(a) If \(bb = b\), prove that \(b = 1_R\). [Hint: Let \(u\) be the solution of \(b \cdot x = 1_R\) and note that \(b \cdot u = b^2 \cdot u\).]

(b) If \(u\) is the solution of the equation \(ax = 1_R\), prove that \(u\) is also a solution of the equation \(xa = 1_R\). [Hint: Use part (a) with \(b = u \cdot a\).]

Exercise 3.1.43

In the ring \( M(\mathbb{C}) \), let

The product of a real number and a matrix is the matrix given by this rule:

The set \( H \) of real quaternions consists of all matrices of the form

(a) Prove that

(b) Show that \( H \) is a noncommutative ring with identity.

(c) Show that \( H \) is a division ring (defined in Exercise 3.1.42). [Hint: If \( M = aI + bi + cj + dk \), then verify that the solution of the equation \( Mx = 1 \) is the matrix \( aI - bi - cj - dk \), where \( r = 1 / (a^2 + b^2 + c^2 + d^2) \).]

(d) Show that the equation \( x^2 = -1 \) has infinitely many solutions in \( H \). [Hint: Consider quaternions of the form \( 0I + bi + cj + dk \), where \( b^2 + c^2 + d^2 = 1 \).]

Exercise 3.1.44

Let \( S \) be a set and let \( P(S) \) be the set of all subsets of \( S \). Define addition and multiplication in \( P(S) \) by the rules

(a) Prove that \( P(S) \) is a commutative ring with identity. [The verification of additive associativity and distributivity is a bit messy, but an informal discussion using Venn diagrams is adequate for appreciating this example. See Exercise 3.1.19 for a special case.]

(b) Show that every element of \( P(S) \) satisfies the equations \( x^2 = x \) and \( x + x = 0_{P(S)} \).

Exercise 3.1.45

Let \( C \) be the set \( \mathbb{R} \times \mathbb{R} \) with the usual coordinatewise addition (as in Theorem 3.1) and a new multiplication given by

Show that with these operations \( C \) is a field.

Exercise 3.1.46

Let \(r\) and \(s\) be positive integers such that \(r\) divides \(ks + 1\) for some \(k\) with \(1 \leq k < r\). Prove that the subset \(\{0, r, 2r, 3r, \dots, (s - 1)r\}\) of \(\mathbb{Z}_s\) is a ring with identity \(ks + 1\) under the usual addition and multiplication in \(\mathbb{Z}_s\). Exercise 3.1.21 is a special case of this result.