Ring Homomorphisms

Prove Theorem 15.1.

Theorem 15.1 (Properties of Ring Homomorphisms): Let $\phi$ be a ring homomorphism from a ring $R$ to a ring $S$. Let $A$ be a subring of $R$ and let $B$ be an ideal of $S$.

1. For any $r \in R$ and any positive integer $n$, $\phi (nr) = n\phi (r)$ and $\phi (r^n) = (\phi (r))^n$.

2. $\phi (A) = \left\{ \phi (a) \mid a \in A\right\}$ is a subring of $S$.

3. If $A$ is an ideal and $\phi$ is onto $S$, then $\phi (A)$ is an ideal.

4. $\phi^{-1}(B) = \left\{ r \in R \mid \phi (r) \in B\right\}$ is an ideal of $R$.

5. If $R$ is commutative, then $\phi (R)$ is commutative.

6. If $R$ has a unity 1, $S \neq \left\{ 0\right\}$, and $\phi$ is onto, then $\phi (1)$ is the unity of $S$ and units in $R$ map to units in $S$.

7. $\phi$ is an isomorphism if and only if $\phi$ is onto and $\ker \phi = \left\{ r \in R \mid \phi (r) = 0\right\} = \left\{ 0\right\}$.

8. If $\phi$ is an isomorphism from $R$ onto $S$, then $\phi^{-1}$ is an isomorphism from $S$ onto $R$.

Prove Theorem 15.2.

Prove Theorem 15.3.

Theorem 15.3 (First Isomorphism Theorem for Rings): Let $\phi$ be a ring homomorphism from $R$ to $S$. Then the mapping from $R/\ker \phi$ to $\phi (R)$, given by $r + \ker \phi \to \phi (r)$, is an isomorphism. In symbols, $R/\ker \phi \approx \phi (R)$.

Prove Theorem 15.4.

Show that the correspondence $x \to 5x$ from $Z_5$ to $Z_{10}$ does not preserve addition.

Show that the correspondence $x \to 3x$ from $Z_4$ to $Z_{12}$ does not preserve multiplication.

Show that the mapping $\phi : D \to F$ in the proof of Theorem 15.6 is a ring homomorphism.

Theorem 15.6 (Field of Quotients): Let $D$ be an integral domain. Then there exists a field $F$ (called the field of quotients of $D$) that contains a subring isomorphic to $D$.

In the proof, $S = \left\{ (a,b) \mid a, b \in D, b \neq 0\right\}$, with the equivalence relation $(a,b) \equiv (c,d)$ if $ad = bc$; $F$ is the set of equivalence classes of $S$ under $\equiv$, and $x/y$ denotes the equivalence class containing $(x,y)$. Addition and multiplication on $F$ are defined by $a/b + c/d = (ad+bc)/(bd)$ and $a/b \cdot c/d = (ac)/(bd)$. The mapping $\phi : D \to F$ is given by $\phi (x) = x/1$.

Prove that every ring homomorphism $\phi$ from $Z_n$ to itself has the form $\phi (x) = ax$, 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.

Prove that the ring $R$ in Example 10 in Chapter 14 is isomorphic to the ring $S = \left\{ \begin{bmatrix} a_1 & a_2 \\ a_3 & a_4 \end{bmatrix} \mid a_i \in Z_2\right\}$.

(Example 10 of Chapter 14: let $R = \left\{ \begin{bmatrix} a_1 & a_2 \\ a_3 & a_4 \end{bmatrix} \mid a_i \in Z\right\}$ and let $I$ be the subset of $R$ consisting of matrices with even entries. Then $I$ is an ideal of $R$, and the factor ring $R/I$ has 16 elements.)

For the ring $R$ in Example 11, determine the number of group isomorphisms and the number of ring isomorphisms from $Z_5$ to $R$.

(Example 11: let $R = \left\{ 0,2,4,6,8\right\}$ under addition and multiplication modulo 10. Note that 6 is the unity and, because $5x = 0$ for all $x$, the characteristic of $R$ is 5. So, the mapping $\phi (n) = n \cdot 6 = 6n$ is an isomorphism with $\ker \phi = \left\langle 5\right\rangle$. Thus $R \approx Z/ \left\langle 5\right\rangle \approx Z_5$ and $R$ is a field.)

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 $Z_3[i] = \left\{ a+bi \mid a,b \in Z_3\right\}$ (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

Show that $\phi : \mathbb {C} \to S$ given by

is a ring isomorphism.

Let $Z[\sqrt{2}] = \left\{ a+b\sqrt{2} \mid a,b \in Z\right\}$ and

Show that $Z[\sqrt{2}]$ and $H$ are isomorphic as rings.

Consider the mapping from $M_2(Z)$ into $Z$ given by

Prove or disprove that this is a ring homomorphism.

Let $R = \left\{ \begin{bmatrix} a & b \\ 0 & c \end{bmatrix} \mid a, b, c \in Z\right\}$. Prove or disprove that the mapping $\begin{bmatrix} a & b \\ 0 & c \end{bmatrix} \to a$ is a ring homomorphism.

Is the mapping from $Z_5$ to $Z_{30}$ given by $x \to 6x$ 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 $x \to 2x$ a ring homomorphism?

Describe the kernel of the homomorphism given in Example 3.

(Example 3: let $\mathbb {R}^{x}$ denote the ring of all polynomials with real coefficients. The mapping $f(x) \to f(1)$ is a ring homomorphism from $\mathbb {R}^{x}$ onto $\mathbb {R}$.)

To which familiar ring are the ones in Exercise 47 in Chapter 12 isomorphic?

(Exercise 47 of Chapter 12: $R = \left\{ \begin{bmatrix} 0 & a \\ 0 & 0 \end{bmatrix} \mid a \in Z\right\}$ and $S = \left\{ \begin{bmatrix} 0 & 0 \\ a & 0 \end{bmatrix} \mid a \in Z\right\}$, subrings of $M_2(Z)$ in which the product of any two elements is zero.)

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 $Z_{25}$ to $Z_{20}$.

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 $A = \left\langle 2\right\rangle$ and $B = \left\langle 8\right\rangle$. 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 $Z_4$.

Prove that $\mathbb {R}^{x}/ \left\langle x^2\right\rangle$ is ring isomorphic to $\left\{ \begin{bmatrix} a & b \\ 0 & a \end{bmatrix} \mid a, b \in \mathbb {R}\right\}$.

Prove that the ring $Z_3[x]/ \left\langle x^2+1\right\rangle$ is isomorphic to the field $Z_3[i]$.

Given that $\phi$ is a ring homomorphism from $Z \oplus Z$ to $Z \oplus Z$ with $\phi ((1,0)) = (0,1)$ and $\phi ((1,1)) = (1,1)$ find a formula for $\phi ((a,b))$.

Show that $(Z \oplus Z)/( \left\langle a\right\rangle \oplus \left\langle b\right\rangle )$ 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.

Let $n$ be an integer with decimal representation $a_ka_{k-1}\cdots a_1a_0$. Prove that $n$ is divisible by 11 if and only if $a_0-a_1+a_2-\cdots (-1)^ka_k$ 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 $Z_m$ to $Z_n$ given by $\phi (x) = x \bmod n$ is a ring homomorphism if and only if $n$ divides $m$.

Let $n$ be an integer with decimal representation $a_ka_{k-1}\cdots a_1a_0$. 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.

Let $n$ be an integer with decimal representation $a_ka_{k-1}\cdots a_1a_0$. Prove that $n$ is divisible by 4 if and only if $a_1a_0$ is divisible by 4.

For any integer $n > 1$, prove that $Z_n[x]/ \left\langle x\right\rangle$ is isomorphic to $Z_n$.

For any integer $n > 1$, prove that $\left\langle x\right\rangle$ 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 $(2 \cdot 10^{75}+2)^{100} \bmod 3$ and $(10^{100}+1)^{99} \bmod 3$.

Determine all ring homomorphisms from $\mathbb {Q}$ to $\mathbb {Q}$.

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 $x \to x^p$ is a ring homomorphism from $R$ to $R$. If $R$ is a field prove that the mapping is an isomorphism.

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 commutative ring with unity is called a principal ideal ring if every ideal of the ring has the form $\left\langle a\right\rangle$. Show that the homomorphic image of a principal ideal ring is a principal ideal ring.

Let $R$ and $S$ be rings.

Let $R$ and $S$ be commutative rings. If $I$ is a maximal ideal of $R$, prove that $I \oplus S$ is a maximal ideal of $R \oplus S$.

Show that $Z_{mn}$ is ring-isomorphic to $Z_m \oplus Z_n$ when $m$ and $n$ are relatively prime.

Show that if $m$ and $n$ are distinct positive integers, then $mZ$ is not ring-isomorphic to $nZ$.

If $\phi$ is a ring automorphism of the field $\mathbb {Q}^{\sqrt[3]{2}} = \left\{ r+s\sqrt[3]{2}+t(\sqrt[3]{2})^2 \mid r,s,t \in \mathbb {Q}\right\}$ determine $\phi (\sqrt[3]{2})$.

Show that the only ring automorphism of the real numbers is the identity mapping.

Determine all ring homomorphisms from $\mathbb {R}$ to $\mathbb {R}$.

Suppose that $n$ divides $m$ and that $a$ is an idempotent of $Z_n$ (i.e., $a^2 = a$). Show that the mapping $x \to ax$ 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 $\mathbb {Q}^{\sqrt{2}} = \left\{ a+b\sqrt{2} \mid a,b \in \mathbb {Q}\right\}$ and $\mathbb {Q}^{\sqrt{5}} = \left\{ a+b\sqrt{5} \mid a,b \in \mathbb {Q}\right\}$. Show that these two rings are not ring-isomorphic.

Let $Z[i] = \left\{ a+bi \mid a,b \in Z\right\}$. Show that the field of quotients of $Z[i]$ is ring-isomorphic to $\mathbb {Q}^{i} = \left\{ r+si \mid r,s \in \mathbb {Q}\right\}$. (This exercise is referred to in Chapter 18.)

Let $F$ be a field. Show that the field of quotients of $F$ is ring-isomorphic 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.

(Corollary 3, A Field Contains $Z_p$ or $\mathbb {Q}$: if $F$ is a field of characteristic $p$, then $F$ contains a subfield isomorphic to $Z_p$. If $F$ is a field of characteristic 0, then $F$ contains a subfield isomorphic to the rational numbers. In the proof of the characteristic-0 case, $F$ has a subring $S$ isomorphic to $Z$, and $T = \left\{ ab^{-1} \mid a,b \in S, b \neq 0\right\}$.)

For any nonzero complex number $a+bi$ show that $Z[i]/ \left\langle a+bi\right\rangle$ is ring-isomorphic to $Z[i]/ \left\langle a-bi\right\rangle$.

Let $f(x) \in \mathbb {R}^{x}$. If $a+bi$ is a complex zero of $f(x)$ (here $i = \sqrt{-1}$), show that $a-bi$ is a zero of $f(x)$. (This exercise is referred to in Chapter 30.)

Let $R = \left\{ \begin{bmatrix} a & b \\ b & a \end{bmatrix} \mid a, b \in Z\right\}$, and let $\phi$ be the mapping that takes $\begin{bmatrix} a & b \\ b & a \end{bmatrix}$ to $a - b$.

Show that the prime subfield of a field of characteristic $p$ is ring-isomorphic to $Z_p$ and that the prime subfield of a field of characteristic 0 is ring-isomorphic to $\mathbb {Q}$. (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 $Z_{2n}$ if and only if $n$ is odd.