§6.1 Exercises

Exercise 6.1.1

Show that the set \(K\) of all constant polynomials in \(\mathbb{Z}[x]\) is a subring but not an ideal in \(\mathbb{Z}[x]\).

Exercise 6.1.2

Show that the set \(I\) of all polynomials with even constant terms is an ideal in \(\mathbb{Z}[x]\).

Exercise 6.1.3

(a) Show that the set \(I = \{(k, 0) \mid k \in \mathbb{Z}\}\) is an ideal in the ring \(\mathbb{Z} \times \mathbb{Z}\).

(b) Show that the set \(T = \{(k, k) \mid k \in \mathbb{Z}\}\) is not an ideal in \(\mathbb{Z} \times \mathbb{Z}\).

Exercise 6.1.4

Is the set \(J = \left\{ \begin{pmatrix} 0 & 0 \\ r & 0 \end{pmatrix} \mid r \in \mathbb{R} \right\}\) an ideal in the ring \(M(\mathbb{R})\) of \(2 \times 2\) matrices over \(\mathbb{R}\)?

Exercise 6.1.5

Show that the set \(K = \left\{ \begin{pmatrix} a & b \\ 0 & 0 \end{pmatrix} \mid a, b \in \mathbb{R} \right\}\) is a subring of \(M(\mathbb{R})\) that absorbs products on the right. Show that \(K\) is not an ideal because it may fail to absorb products on the left. Such a set \(K\) is sometimes called a right ideal.

Exercise 6.1.6

(a) Show that the set of nonunits in \(\mathbb{Z}_8\) is an ideal.

(b) Do part (a) for \(\mathbb{Z}_9\). [Also, see Exercise 24.]

Exercise 6.1.7

Let \(c \in R\) and let \(I = \{rc \mid r \in R\}\).

(a) If \(R\) is commutative, prove that \(I\) is an ideal (that is, Theorem 6.2 is true even when \(R\) does not have an identity).

(b) If \(R\) is commutative but has no identity, is \(c\) an element of the ideal \(I\)?

[Hint: Consider the ideal \(\{2k \mid k \in E\}\) in the ring \(E\) of even integers. Also see Exercise 33.]

(c) Give an example to show that if \(R\) is not commutative, then \(I\) need not be an ideal.

Exercise 6.1.8

If \(I\) is an ideal in \(R\) and \(J\) is an ideal in the ring \(S\), prove that \(I \times J\) is an ideal in the ring \(R \times S\).

Exercise 6.1.9

Let \(R\) be a ring with identity and let \(I\) be an ideal in \(R\).

(a) If \(1_R \in I\), prove that \(I = R\).

(b) If \(I\) contains a unit, prove that \(I = R\).

Exercise 6.1.10

If \(I\) is an ideal in a field \(F\), prove that \(I = (0_F)\) or \(I = F\).

[Hint: Exercise 9.]

Exercise 6.1.11

List the distinct principal ideals in each ring:

(a) \(\mathbb{Z}_5\)

(b) \(\mathbb{Z}_9\)

(c) \(\mathbb{Z}_{12}\)

Exercise 6.1.12

List the distinct principal ideals in \(\mathbb{Z}_2 \times \mathbb{Z}_3\).

Exercise 6.1.13

If \(R\) is a commutative ring with identity and \((a)\) and \((b)\) are principal ideals such that \((a) = (b)\), is it true that \(a = b\)? Justify your answer.

Exercise 6.1.14

Prove Theorem 6.3.

Exercise 6.1.15

Show that the ideal generated by \(x\) and 2 in the ring \(\mathbb{Z}[x]\) is the ideal \(I\) of all polynomials with even constant terms (see Example 9).

Exercise 6.1.16

(a) Show that \((4, 6) = (2)\) in \(\mathbb{Z}\), where \((4, 6)\) is the ideal generated by 4 and 6 and \((2)\) is the principal ideal generated by 2.

(b) Show that \((6, 9, 15) = (3)\) in \(\mathbb{Z}\).

Exercise 6.1.17

(a) If \(I\) and \(J\) are ideals in \(R\), prove that \(I \cap J\) is an ideal.

(b) If \(\{I_k\}\) is a (possibly infinite) family of ideals in \(R\), prove that the intersection of all the \(I_k\) is an ideal.

Exercise 6.1.18

Give an example in \(\mathbb{Z}\) to show that the set theoretic union of two ideals may not be an ideal (in fact, it may not even be a subring).

Exercise 6.1.19

If \(I\) is an ideal in \(R\) and \(S\) is a subring of \(R\), prove that \(I \cap S\) is an ideal in \(S\).

Exercise 6.1.20

Let \(I\) and \(J\) be ideals in \(R\). Prove that the set \(K = \{a + b \mid a \in I, b \in J\}\) is an ideal in \(R\) that contains both \(I\) and \(J\). \(K\) is called the sum of \(I\) and \(J\) and is denoted \(I + J\).

Exercise 6.1.21

If \(d\) is the greatest common divisor of \(a\) and \(b\) in \(\mathbb{Z}\), show that \((a) + (b) = (d)\).

(The sum of ideals is defined in Exercise 20.)

Exercise 6.1.22

Let \(I\) and \(J\) be ideals in \(R\). Is the set \(K = \{ab \mid a \in I, b \in J\}\) an ideal in \(R\)? Compare Exercise 20.

Exercise 6.1.23

(a) Verify that \(I = \{0, 3\}\) is an ideal in \(\mathbb{Z}_6\) and list all its distinct cosets.

(b) Verify that \(I = \{0, 3, 6, 9, 12\}\) is an ideal in \(\mathbb{Z}_{15}\) and list all its distinct cosets.

Exercise 6.1.24

Let \(R\) be a commutative ring with identity, and let \(N\) be the set of nonunits in \(R\). Give an example to show that \(N\) need not be an ideal.

Exercise 6.1.25

Let \(J\) be an ideal in \(R\). Prove that \(I\) is an ideal, where:

\[ I = \{r \in R \mid rJ = 0_R \ \text{for every} \ r \in J \}. \]

Exercise 6.1.26

Let \(J\) be an ideal in \(R\). Prove that \(K\) is an ideal, where:

\[ K = \{a \in R \mid ra \in I \ \text{for every} \ r \in R\}. \]

Exercise 6.1.27

Let \(f: R \rightarrow S\) be a homomorphism of rings and let:

\[ K = \{r \in R \mid f(r) = 0_S\}. \]

Prove that \(K\) is an ideal in \(R\).

Exercise 6.1.28

If \(I\) is an ideal in \(R\), prove that \(I[x]\) (polynomials with coefficients in \(I\)) is an ideal in the polynomial ring \(R[x]\).

Exercise 6.1.29

If \((m, n) = 1\) in \(\mathbb{Z}\), prove that \((m) \cap (n)\) is the ideal \((mn)\).

Exercise 6.1.30

Prove that the set of nilpotent elements in a commutative ring \(R\) is an ideal.

[Hint: See Exercise 44 in Section 3.2.]

Exercise 6.1.31

Let \(R\) be an integral domain and \(a, b \in R\). Show that \((a) = (b)\) if and only if \(a = bu\) for some unit \(u \in R\).

Exercise 6.1.32

(a) Prove that the set \(J\) of all polynomials in \(\mathbb{Z}[x]\) whose constant terms are divisible by 3 is an ideal.

(b) Show that \(J\) is not a principal ideal.

Exercise 6.1.33

Let \(R\) be a commutative ring without identity and let \(a \in R\). Show that:

\[ A = \{ra + na \mid r \in R, n \in \mathbb{Z}\} \]

is an ideal containing \(a\) and that every ideal containing \(a\) also contains \(A\). \(A\) is called the principal ideal generated by \(a\).

Exercise 6.1.34

If \(M\) is an ideal in a commutative ring \(R\) with identity and if \(a \in R\) with \(a \notin M\), prove that the set:

\[ J = \{m + ra \mid r \in R \ \text{and} \ m \in M\} \]

is an ideal such that \(M \subset J\).

Exercise 6.1.35

Let \(I\) be an ideal in \(\mathbb{Z}\) such that \((3) \subset I \subset \mathbb{Z}\). Prove that either \(I = (3)\) or \(I = \mathbb{Z}\).

Exercise 6.1.36

Let \(I\) and \(J\) be ideals in \(R\). Let \(IJ\) denote the set of all possible finite sums of elements of the form \(ab\) (with \(a \in I\), \(b \in J\)), that is:

\[ IJ = \{a_1b_1 + a_2b_2 + \cdots + a_n b_n \mid n \geq 1, a_k \in I, b_k \in J\}. \]

Prove that \(IJ\) is an ideal. \(IJ\) is called the product of \(I\) and \(J\).

Exercise 6.1.37

Let \(R\) be a commutative ring with identity \(1_R \neq 0_R\) whose only ideals are \((0_R)\) and \(R\). Prove that \(R\) is a field.

[Hint: If \(a \neq 0_R\), use the ideal \((a)\) to find a multiplicative inverse for \(a\).]

Exercise 6.1.38

Let \(J\) be an ideal in a commutative ring \(R\) and let:

\[ J = \{r \in R \mid r^n \in I \ \text{for some positive integer} \ n\}. \]

Exercise 6.1.39

(a) Show that the ring \(M(\mathbb{R})\) is not a division ring by exhibiting a matrix that has no multiplicative inverse. (Division rings are defined in Exercise 42 of Section 3.1.)

(b) Show that \(M(\mathbb{R})\) has no ideals except the zero ideal and \(M(\mathbb{R})\) itself.

[Hint: If \(J\) is a nonzero ideal, show that \(J\) contains a matrix \(A\) with a nonzero entry in the upper left-hand corner. Verify that:

\[ \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} \cdot A \cdot \begin{pmatrix} C^{-1} & 0 \\ 0 & 0 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} \]

and that this matrix is in \(J\). Similarly, show that:

\[ \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} \]

is in \(J\). What is their sum? See Exercise 9.]

Exercise 6.1.40

Prove that every ideal in \(\mathbb{Z}\) is principal.

[Hint: If \(I\) is a nonzero ideal, show that \(I\) must contain positive elements and, hence, must contain a smallest positive element \(c\) (why?). Since \(c \in I\), every multiple of \(c\) is also in \(I\), hence \((c) \subseteq I\). To show that \(I \subseteq (c)\), let \(a\) be any element of \(I\). Then \(a = cq + r\) with \(0 \leq r < c\). Show that \(r=0\), so that \(a=cq \in (c)\].

Exercise 6.1.41

(a) Prove that the set \(S\) of rational numbers (in lowest terms) with odd denominators is a subring of \(\mathbb{Q}\).

(b) Let \(I\) be the set of elements of \(S\) with even numerators. Prove that \(I\) is an ideal in \(S\).

(c) Show that \(S/I\) consists of exactly two distinct cosets.

Exercise 6.1.42

(a) Let \(p\) be a prime integer and let \(T\) be the set of rational numbers (in lowest terms) whose denominators are not divisible by \(p\). Prove that \(T\) is a ring.

(b) Let \(I\) be the set of elements of \(T\) whose numerators are divisible by \(p\). Prove that \(I\) is an ideal in \(T\).

(c) Show that \(T/I\) consists of exactly \(p\) distinct cosets.

Exercise 6.1.43

Let \(J\) be the set of all polynomials with zero constant term in \(\mathbb{Z}[x]\).

(a) Show that \(J\) is the principal ideal \((x)\) in \(\mathbb{Z}[x]\).

(b) Show that \(\mathbb{Z}[x]/J\) consists of an infinite number of distinct cosets, one for each \(n \in \mathbb{Z}\).

Exercise 6.1.44

(a) Prove that the set \(T\) of matrices of the form \(\begin{pmatrix} a & b \\ 0 & a \end{pmatrix}\) with \(a, b \in \mathbb{R}\) is a subring of \(M(\mathbb{R})\).

(b) Prove that the set \(I\) of matrices of the form \(\begin{pmatrix} 0 & b \\ 0 & 0 \end{pmatrix}\) with \(b \in \mathbb{R}\) is an ideal in the ring \(T\).

(c) Show that every coset in \(T/I\) can be written in the form \(\begin{pmatrix} a & 0 \\ 0 & a \end{pmatrix} + I\).

Exercise 6.1.45

(a) Prove that the set \(S\) of matrices of the form \(\begin{pmatrix} a & 0 \\ c & b \end{pmatrix}\) with \(a, b, c \in \mathbb{R}\) is a subring of \(M(\mathbb{R})\).

(b) Prove that the set \(I\) of matrices of the form \(\begin{pmatrix} 0 & 0 \\ b & 0 \end{pmatrix}\) with \(b \in \mathbb{R}\) is an ideal in the ring \(S\).

(c) Show that there are infinitely many distinct cosets in \(S/I\), one for each pair in \(\mathbb{R} \times \mathbb{R}\).

Exercise 6.1.46

Let \(F\) be a field. Prove that every ideal in \(F[x]\) is principal.

[Hint: Use the Division Algorithm to show that the nonzero ideal \(I\) in \(F[x]\) is \((p(x))\), where \(p(x)\) is a polynomial of smallest possible degree in \(I)\].

Exercise 6.1.47

Prove that a subring \(S\) of \(\mathbb{Z}_n\) has an identity if and only if there is an element \(u\) in \(S\) such that \(u^2 = u\) and \(S\) is the ideal \((u)\).