Integral Domains

Verify that Examples 1 through 8 are as claimed.

Example 1: The ring of integers is an integral domain. Example 2: The ring of Gaussian integers $Z[i] = \left\{ a+bi \mid a, b \in Z\right\}$ is an integral domain. Example 3: The ring $Z[x]$ of polynomials with integer coefficients is an integral domain. Example 4: The ring $Z[\sqrt{2}] = \left\{ a+b\sqrt{2} \mid a, b \in Z\right\}$ is an integral domain. Example 5: The ring $Z_p$ of integers modulo a prime $p$ is an integral domain. Example 6: The ring $Z_n$ of integers modulo $n$ is not an integral domain when $n$ is not prime. Example 7: The ring $M_2(Z)$ of $2 \times 2$ matrices over the integers is not an integral domain. Example 8: $Z \oplus Z$ is not an integral domain.

Which of Examples 1 through 5 are fields?

Show that a commutative ring with the cancellation property (under multiplication) has no zero-divisors.

List all zero-divisors in $Z_{20}$. Can you see a relationship between the zero-divisors of $Z_{20}$ and the units of $Z_{20}$?

Show that every nonzero element of $Z_n$ is a unit or a zero-divisor.

Find a nonzero element in a ring that is neither a zero-divisor nor a unit.

Let $R$ be a finite commutative ring with unity. Prove that every nonzero element of $R$ is either a zero-divisor or a unit. What happens if we drop the ``finite'' condition on $R$?

Let $a \neq 0$ belong to a commutative ring. Prove that $a$ is a zero-divisor if and only if $a^2 b = 0$ for some $b \neq 0$.

Find elements $a$, $b$, and $c$ in the ring $Z \oplus Z \oplus Z$ such that $ab$, $ac$, and $bc$ are zero-divisors but $abc$ is not a zero-divisor.

Describe all zero-divisors and units of $Z \oplus Q \oplus Z$.

Let $d$ be an integer. Prove that $Z[\sqrt{d}] = \left\{ a+b\sqrt{d} \mid a, b \in Z\right\}$ is an integral domain. (This exercise is referred to in Chapter 18.)

In $Z_7$, give a reasonable interpretation for the expressions $1/2$, $-2/3$, $\sqrt{-3}$, and $-1/6$.

Give an example of a commutative ring without zero-divisors that is not an integral domain.

Find two elements $a$ and $b$ in a ring such that both $a$ and $b$ are zero-divisors, $a + b \neq 0$, and $a + b$ is not a zero-divisor.

Let $a$ belong to a ring $R$ with unity and suppose that $a^n = 0$ for some positive integer $n$. (Such an element is called nilpotent.) Prove that $1 - a$ has a multiplicative inverse in $R$.

Show that the nilpotent elements of a commutative ring form a subring.

Show that 0 is the only nilpotent element in an integral domain.

A ring element $a$ is called an idempotent if $a^2 = a$. Prove that the only idempotents in an integral domain are 0 and 1.

Let $a$ and $b$ be idempotents in a commutative ring. Show that each of the following is also an idempotent: $ab$, $a - ab$, $a + b - ab$, $a + b - 2ab$. If $a$ is an element in a ring with the property that $a^2 = -a$, what can you say about $-a$?

Show that $Z_n$ has a nonzero nilpotent element if and only if $n$ is divisible by the square of some prime.

Let $R$ be the ring of real-valued continuous functions on $[-1,1]$. Show that $R$ has zero-divisors.

Prove that if $a$ is a ring idempotent, then $a^n = a$ for all positive integers $n$.

Determine all ring elements that are both nilpotent elements and idempotents.

Find an idempotent in $Z_5[i] = \left\{ a+bi \mid a,b \in Z_5\right\}$.

Use Table 13.1 to find a generator for the cyclic group $Z_3[i]^*$.

Find all units, zero-divisors, idempotents, and nilpotent elements in $Z_3 \oplus Z_6$.

Determine all elements of a ring that are both units and idempotents.

Let $R$ be the set of all real-valued functions defined for all real numbers under function addition and multiplication.

(Subfield Test) Let $F$ be a field and let $K$ be a subset of $F$ with at least two elements. Prove that $K$ is a subfield of $F$ if, for any $a, b$ ($b \neq 0$) in $K$, $a - b$ and $ab^{-1}$ belong to $K$.

Let $d$ be a positive integer. Prove that $Q[\sqrt{d}] = \left\{ a+b\sqrt{d} \mid a,b \in Q\right\}$ is a field.

Let $R$ be a ring with unity 1. If the product of any pair of nonzero elements of $R$ is nonzero, prove that $ab = 1$ implies $ba = 1$.

Let $R = \left\{ 0,2,4,6,8\right\}$ under addition and multiplication modulo 10. Prove that $R$ is a field.

Formulate the appropriate definition of a subdomain (i.e., a ``sub'' integral domain). Let $D$ be an integral domain with unity 1. Show that $P = \left\{ n \cdot 1 \mid n \in Z\right\}$ (i.e., all integral multiples of 1) is a subdomain of $D$. Show that $P$ is contained in every subdomain of $D$. What can we say about the order of $P$?

Prove that there is no integral domain with exactly six elements. Can your argument be adapted to show that there is no integral domain with exactly four elements? What about 15 elements? Use these observations to guess a general result about the number of elements in a finite integral domain.

Let $F$ be a field of order $2^n$. Prove that $\operatorname {char} F = 2$. Does your proof remain valid when 2 is replaced by any prime $p$?

Determine all elements of an integral domain that are their own inverses under multiplication.

Characterize those integral domains for which 1 is the only element that is its own multiplicative inverse.

Determine all integers $n > 1$ for which $(n-1)!$ is a zero-divisor in $Z_n$.

Suppose that $a$ and $b$ belong to an integral domain.

Find an example of an integral domain and distinct positive integers $m$ and $n$ such that $a^m = b^m$ and $a^n = b^n$, but $a \neq b$.

Let $F$ be a field of order 32. Show that the only subfields of $F$ are $F$ itself and $\left\{ 0,1\right\}$.

Construct a multiplication table for $Z_2[i]$, the ring of Gaussian integers modulo 2. Is this ring a field? Is it an integral domain?

Show that $Z_7[\sqrt{3}] = \left\{ a+b\sqrt{3} \mid a,b \in Z_7\right\}$ is a field. For any positive integer $k$ and any prime $p$, determine a necessary and sufficient condition for $Z_p[\sqrt{k}] = \left\{ a+b\sqrt{k} \mid a,b \in Z_p\right\}$ to be a field.

The nonzero elements of $Z_3[i]$ form an Abelian group of order 8 under multiplication. Is it isomorphic to $Z_8$, $Z_4 \oplus Z_2$, or $Z_2 \oplus Z_2 \oplus Z_2$?

Show that a finite commutative ring with no zero-divisors and at least two elements has a unity.

Suppose that $a$ and $b$ belong to a commutative ring and $ab$ is a zero-divisor. Show that either $a$ or $b$ is a zero-divisor.

Suppose that $R$ is a commutative ring without zero-divisors. Show that all the nonzero elements of $R$ have the same additive order.

Suppose that $R$ is a commutative ring without zero-divisors. Show that the characteristic of $R$ is 0 or prime.

Let $x_1, x_2, \ldots , x_n$ belong to a commutative ring with prime characteristic $p$.

Let $R$ be a commutative ring with unity 1 and prime characteristic. If $a \in R$ is nilpotent, prove that there is a positive integer $k$ such that $(1+a)^k = 1$.

Show that any finite field has order $p^n$, where $p$ is a prime.

Give an example of an infinite integral domain that has characteristic 3.

Let $R$ be a ring and let $M_2(R)$ be the ring of $2 \times 2$ matrices with entries from $R$. Explain why these two rings have the same characteristic.

Let $R$ be a ring with $m$ elements. Show that the characteristic of $R$ divides $m$.

Explain why a finite ring must have a nonzero characteristic.

Find all solutions of $x^2 - x + 2 = 0$ over $Z_3[i]$. (See Example 9.)

Consider the equation $x^2 - 5x + 6 = 0$.

If $R$ is a ring of characteristic $m > 0$ and $S$ is a subring of $R$, what can you say about the characteristic of $S$?

Suppose that $R$ is an integral domain in which $20 \cdot 1 = 0$ and $12 \cdot 1 = 0$. (Recall that $n \cdot 1$ means the sum $1+1+\cdots +1$ with $n$ terms.) What is the characteristic of $R$?

In a commutative ring of characteristic 2, prove that the idempotents form a subring.

For a prime $p$ prove that if $x^2 + 1 = 0$ has a solution in $Z_p$, then $Z_p[i] = \left\{ a+bi \mid a,b \in Z_p\right\}$ is not a field.

Prove that any idempotent in a commutative ring with unity other than 0 or 1 is a zero divisor.

Show that any field of order 16 has no subfield of order 8.

Let $F$ be a finite field with $n$ elements. Prove that $x^{n-1} = 1$ for all nonzero $x$ in $F$.

Let $F$ be a field of prime characteristic $p$. Prove that $K = \left\{ x \in F \mid x^p = x\right\}$ is a subfield of $F$.

Prove that the set of complex numbers $Q(i) = \left\{ a+bi \mid a,b \in Q\right\}$ is a field. This field is called the Gaussian rationals.

Let $R = Z_2 \oplus Z_3 \oplus Z_4 \oplus \cdots$. What is the identity of $R$? What is $\operatorname {char} R$? Prove that for each positive integer $n$, $R$ has a subring that has characteristic $n$.

If $a$ is an idempotent in a commutative ring $R$ prove that $a$ is the identity in the ideal $\left\langle a\right\rangle$.

Let $S$ be a subring of a ring $R$ and suppose that $u_S$ is a unity in $S$ and $u_R$ is a unity in $R$ and $u_S \neq u_R$. Prove that $u_S$ and $u_R - u_S$ are zero divisors in $R$.

If $a, b$ belong to an integral domain prove that $\left\langle a\right\rangle = \left\langle b\right\rangle$ if and only if $a = bu$ where $u$ is a unit.

Suppose that $F$ is a field with 27 elements. Show that for every element $a \in F$, $5a = -a$.

Suppose that $a$ and $b$ belong to a field of order 8 and that $a^2 + ab + b^2 = 0$. Prove that $a = 0$ and $b = 0$. Do the same when the field has order $2^n$ with $n$ odd.

Let $F$ be a field of characteristic 2 with more than two elements. Show that $(x+y)^3 \neq x^3+y^3$ for some $x$ and $y$ in $F$.