Chapter Questions
Prove Theorem $15.1$.
Prove Theorem $15.2$.
Prove Theorem $15.3$.
Prove Theorem $15.4 .$
Show that the correspondence $x \rightarrow 5 x$ from $Z_{5}$ to $Z_{10}$ does not preserve addition.
Show that the correspondence $x \rightarrow 3 x$ from $Z_{4}$ to $Z_{12}$ does not preserve multiplication.
Show that the mapping $\phi: D \rightarrow F$ in the proof of Theorem $15.6$ is a ring homomorphism.
Prove that every ring homomorphism $\phi$ from $Z_{n}$ to itself has the form $\phi(x)=a x$, where $a^{2}=a$
Suppose that $\phi$ is a ring homomorphism from $Z_{m}$ to $Z_{n} .$ Prove that if $\phi(1)=a$, then $a^{2}=a .$ Give an example to show that the converse is false.
a. Is the ring $2 Z$ isomorphic to the ring $3 Z$ ?b. Is the ring $2 Z$ isomorphic to the ring $4 Z$ ?
Prove that the intersection of any collection of subfields of a field $F$ is a subfield of $F$. (This exercise is referred to in this chapter.)
Let $Z_{3}[i]=\left\{a+b i \mid a, b \in Z_{3}\right\}$ (see Example 9 in Chapter 13). Show that the field $Z_{3}[i]$ is ring-isomorphic to the field $Z_{3}[x] /\left\langle x^{2}+1\right\rangle$.
Let$$S=\left\{\left[\begin{array}{rr}a & b \\-b & a\end{array}\right] \mid a, b \in \mathbf{R}\right\}$$Show that $\phi: \mathbf{C} \rightarrow S$ given by$$\phi(a+b i)=\left[\begin{array}{rr}a & b \\-b & a\end{array}\right]$$is a ring isomorphism.
Let $Z[\sqrt{2}]=\{a+b \sqrt{2} \mid a, b \in Z\}$ and$$H=\left\{\left[\begin{array}{lr}a & 2 b \\b & a\end{array}\right] \mid a, b \in Z\right\} .$$Show that $Z[\sqrt{2}$ ] and $H$ are isomorphic as rings.
Consider the mapping from $M_{2}(Z)$ into $Z$ given by $\left[\begin{array}{cc}a & b \\ c & d\end{array}\right] \rightarrow a$. Prove or disprove that this is a ring homomorphism.
Let $R=\left\{\left[\begin{array}{ll}a & b \\ 0 & c\end{array}\right] \mid a, b, c \in Z\right\} .$ Prove or disprove that the mapping $\left[\begin{array}{cc}a & b \\ 0 & c\end{array}\right] \rightarrow a$ is a ring homomorphism.
Is the mapping from $Z_{5}$ to $Z_{30}$ given by $x \rightarrow 6 x$ a ring homomorphism? Note that the image of the unity is the unity of the image but not the unity of $Z_{30}$.
Is the mapping from $Z_{10}$ to $Z_{10}$ given by $x \rightarrow 2 x$ a ring homomorphism?
Describe the kernel of the homomorphism given in Example $3 .$
Recall that a ring element $a$ is called an idempotent if $a^{2}=a$. Prove that a ring homomorphism carries an idempotent to an idempotent.
Determine all ring homomorphisms from $Z_{6}$ to $Z_{6}$. Determine all ring homomorphisms from $Z_{20}$ to $Z_{30}$.
Determine all ring isomorphisms from $Z_{n}$ to itself.
Determine all ring homomorphisms from $Z$ to $Z$.
Suppose $\phi$ is a ring homomorphism from $Z \oplus Z$ into $Z \oplus Z$. What are the possibilities for $\phi((1,0))$ ?
Determine all ring homomorphisms from $Z \oplus Z$ into $Z \oplus Z$.
In $Z$, let $A=\langle 2\rangle$ and $B=\langle 8\rangle .$ Show that the group $A / B$ is isomorphic to the group $Z_{4}$ but that the ring $A / B$ is not ring-isomorphic to the ring $Z_{4}$.
Let $R$ be a ring with unity and let $\phi$ be a ring homomorphism from $R$ onto $S$ where $S$ has more than one element. Prove that $S$ has a unity.
Show that $(Z \oplus Z) /(\langle a\rangle \oplus\langle b\rangle)$ is ring-isomorphic to $Z_{a} \oplus Z_{b}$.
Determine all ring homomorphisms from $Z \oplus Z$ to $Z$.
Prove that the sum of the squares of three consecutive integers cannot be a square.
Let $m$ be a positive integer and let $n$ be an integer obtained from $m$ by rearranging the digits of $m$ in some way. (For example, 72345 is a rearrangement of $35274 .$ Show that $m-n$ is divisible by 9 .
(Test for Divisibility by 11 ) Let $n$ be an integer with decimal representation $a_{k} a_{k-1} \cdots a_{1} a_{0} .$ Prove that $n$ is divisible by 11 if and only if $a_{0}-a_{1}+a_{2}-\cdots(-1)^{k} a_{k}$ is divisible by 11 .
Show that the number $7,176,825,942,116,027,211$ is divisible by 9 but not divisible by 11 .
If $m$ and $n$ are positive integers, prove that the mapping from $\mathrm{Z}_{m}$ to $\mathrm{Z}_{n}$ given by $\phi(x)=x \bmod n$ is a ring homomorphism if and only if $n$ divides $m$.
(Test for Divisibility by 3 ) Let $n$ be an integer with decimal representation $a_{k} a_{k-1} \cdots a_{1} a_{0} .$ Prove that $n$ is divisible by 3 if and only if $a_{k}+a_{k-1}+\cdots+a_{1}+a_{0}$ is divisible by $3 .$
(Test for Divisibility by 4) Let $n$ be an integer with decimal representation $a_{k} a_{k-1} \cdots a_{1} a_{0} .$ Prove that $n$ is divisible by 4 if and only if $a_{1} a_{0}$ is divisible by $4 .$
For any integer $n>1$, prove that $Z_{n}[x] /\langle x\rangle$ is isomorphic to $Z_{n}$.
For any integer $n>1$, prove that $\langle x\rangle$ is a maximal ideal of $Z_{n}[x]$ if and only if $n$ is prime.
Give an example of a ring homomorphism from a commutative ring $R$ to a ring $S$ that maps a zero-divisor in $R$ to the unity of $S$.
Prove that any automorphism of a field $F$ is the identity from the prime subfield to itself.
In your head, determine $\left(2 \cdot 10^{75}+2\right)^{100} \bmod 3$ and $\left(10^{100}+1\right)^{99} \bmod 3$.
Determine all ring homomorphisms from $Q$ to $Q$.
Let $R$ and $S$ be commutative rings with unity. If $\phi$ is a homomorphism from $R$ onto $S$ and the characteristic of $R$ is nonzero, prove that the characteristic of $S$ divides the characteristic of $R$.
Let $R$ be a commutative ring of prime characteristic $p$. Show that the Frobenius map $x \rightarrow x^{p}$ is a ring homomorphism from $R$ to $R$.
Is there a ring homomorphism from the reals to some ring whose kernel is the integers?
Show that a homomorphism from a field onto a ring with more than one element must be an isomorphism.
Suppose that $R$ and $S$ are commutative rings with unities. Let $\phi$ be a ring homomorphism from $R$ onto $S$ and let $A$ be an ideal of $S$.a. If $A$ is prime in $S$, show that $\phi^{-1}(A)=\{x \in R \mid \phi(x) \in A\}$ is prime in $R$.b. If $A$ is maximal in $S$, show that $\phi^{-1}(A)$ is maximal in $R$.
A principal ideal ring is a ring with the property that every ideal has the form $\langle a\rangle$. Show that the homomorphic image of a principal ideal ring is a principal ideal ring.
Let $R$ and $S$ be rings.a. Show that the mapping from $R \oplus S$ onto $R$ given by $(a, b) \rightarrow a$ is a ring homomorphism.b. Show that the mapping from $R$ to $R \oplus S$ given by $a \rightarrow(a, 0)$ is a one-to-one ring homomorphism.c. Show that $R \oplus S$ is ring-isomorphic to $S \oplus R$.
Show that if $m$ and $n$ are distinct positive integers, then $m Z$ is not ring-isomorphic to $n Z$.
Prove or disprove that the field of real numbers is ring-isomorphic to the field of complex numbers.
Show that the only ring automorphism of the real numbers is the identity mapping.
Determine all ring homomorphisms from $\mathbf{R}$ to $\mathbf{R}$.
Suppose that $n$ divides $m$ and that $a$ is an idempotent of $Z_{n}$ (that is, $a^{2}=a$ ). Show that the mapping $x \rightarrow a x$ is a ring homomorphism from $Z_{m}$ to $Z_{n} .$ Show that the same correspondence need not yield a ring homomorphism if $n$ does not divide $m$.
Show that the operation of multiplication defined in the proof of Theorem $15.6$ is well-defined.
Let $Q[\sqrt{2}]=\{a+b \sqrt{2} \mid a, b \in Q\}$ and $Q[\sqrt{5}]=\{a+b \sqrt{5}$ ? $a, b \in Q\}$. Show that these two rings are not ring-isomorphic.
Let $Z[i]=\{a+b i \mid a, b \in Z\} .$ Show that the field of quotients of $Z[i]$ is ring-isomorphic to $Q[i]=\{r+s i \mid r, s \in Q\} .$ (This exercise is referred to in Chapter $18 .$ )
Let $F$ be a field. Show that the field of quotients of $F$ is ringisomorphic to $F$.
Let $D$ be an integral domain and let $F$ be the field of quotients of $D$. Show that if $E$ is any field that contains $D$, then $E$ contains a subfield that is ring-isomorphic to $F$. (Thus, the field of quotients of an integral domain $D$ is the smallest field containing $D .$ )
Explain why a commutative ring with unity that is not an integral domain cannot be contained in a field. (Compare with Theorem 15.6.)
Show that the relation $\equiv$ defined in the proof of Theorem $15.6$ is an equivalence relation.
Give an example of a ring without unity that is contained in a field.
Prove that the set $T$ in the proof of Corollary 3 to Theorem $15.5$ is ring-isomorphic to the field of rational numbers.
Suppose that $\phi: R \rightarrow S$ is a ring homomorphism and that the image of $\phi$ is not $\{0\}$. If $R$ has a unity and $S$ is an integral domain, show that $\phi$ carries the unity of $R$ to the unity of $S$. Give an example to show that the preceding statement need not be true if $S$ is not an integral domain.
Let $f(x) \in \mathbf{R}[x] .$ If $a+b i$ is a complex zero of $f(x)$ (here $i=\sqrt{-1})$ show that $a-b i$ is a zero of $f(x)$. (This exercise is referred to in Chapter 32.)
Let $R=\left\{\left[\begin{array}{ll}a & b \\ b & a\end{array}\right] \mid a, b \in Z\right\}$, and let $\phi$ be the mapping that takes $\left[\begin{array}{ll}a & b \\ b & a\end{array}\right]$ to $a-b$a. Show that $\phi$ is a homomorphism.b. Determine the kernel of $\phi$.c. Show that $R / \operatorname{Ker} \phi$ is isomorphic to $Z$.d. Is Ker $\phi$ a prime ideal?e. Is Ker $\phi$ a maximal ideal?
Show that the prime subfield of a field of characteristic $p$ is ringisomorphic to $Z_{p}$ and that the prime subfield of a field of characteristic 0 is ring-isomorphic to $Q .$ (This exercise is referred to in this chapter.)
Let $n$ be a positive integer. Show that there is a ring isomorphism from $Z_{2}$ to a subring of $Z_{2 n}$ if and only if $n$ is odd.
Show that $Z_{m n}$ is ring-isomorphic to $Z_{m} \oplus Z_{n}$ when $m$ and $n$ are relatively prime.