Divisibility in Integral Domains

For the ring $\mathbb {Z}^{\sqrt{d}} = \left\{ a+b\sqrt{d} \mid a, b \in \mathbb {Z}\right\}$, where $d \neq 1$ and $d$ is not divisible by the square of a prime, prove that the norm $N(a+b\sqrt{d}) = \left|a^{2}-db^{2}\right|$ satisfies the four assertions made preceding Example 1. (This exercise is referred to in this chapter.)

The four assertions: $N(x) = 0$ if and only if $x = 0$; $N(xy) = N(x)N(y)$ for all $x$ and $y$; $x$ is a unit if and only if $N(x) = 1$; and, if $N(x)$ is prime, then $x$ is irreducible in $\mathbb {Z}^{\sqrt{d}}$.

In an integral domain, show that $a$ and $b$ are associates if and only if $\left\langle a\right\rangle = \left\langle b\right\rangle$.

Show that the union of a chain $I_{1} \subset I_{2} \subset \cdots$ of ideals of a ring $R$ is an ideal of $R$. (This exercise is referred to in this chapter.)

In an integral domain, show that the product of an irreducible and a unit is an irreducible.

Suppose that $a$ and $b$ belong to an integral domain and $b \neq 0$. Show that $\left\langle ab\right\rangle$ is a proper subset of $\left\langle b\right\rangle$ if and only if $a$ is not a unit. This exercise is referred to in this chapter.

Let $D$ be an integral domain. Define $a \sim b$ if $a$ and $b$ are associates. Show that this defines an equivalence relation on $D$.

In the notation of Example 7 (the ring of Gaussian integers $\mathbb {Z}^{i} = \left\{ a+bi \mid a, b \in \mathbb {Z}\right\}$ with $d(a+bi) = a^{2}+b^{2}$), show that $d(xy) = d(x)d(y)$.

Let $D$ be a Euclidean domain with measure $d$. Prove that $u$ is a unit in $D$ if and only if $d(u) = d(1)$.

Let $D$ be a Euclidean domain with measure $d$. Show that if $a$ and $b$ are associates in $D$, then $d(a) = d(b)$.

Let $D$ be a principal ideal domain and let $p \in D$. Prove that $\left\langle p\right\rangle$ is a maximal ideal in $D$ if and only if $p$ is irreducible.

Let $d$ be an integer less than $-1$ that is not divisible by the square of a prime. Prove that the only units of $\mathbb {Z}^{\sqrt{d}}$ are $+1$ and $-1$.

Let $D$ be a principal ideal domain. Show that every proper ideal of $D$ is contained in a maximal ideal of $D$.

In $\mathbb {Z}^{\sqrt{-5}}$, show that $21$ does not factor uniquely as a product of irreducibles.

Show that $1-i$ is an irreducible in $\mathbb {Z}^{i}$.

Show that $\mathbb {Z}^{\sqrt{-6}}$ is not a unique factorization domain. Why does this show that $\mathbb {Z}^{\sqrt{-6}}$ is not a principal ideal domain?

Give an example of a unique factorization domain with a subdomain that does not have a unique factorization.

In $\mathbb {Z}^{i}$, show that $3$ is irreducible but $2$ and $5$ are not.

Prove that $7$ is irreducible in $\mathbb {Z}^{\sqrt{6}}$, even though $N(7)$ is not prime.

Prove that if $p$ is a prime in $\mathbb {Z}$ that can be written in the form $a^{2}+b^{2}$, then $a+bi$ is irreducible in $\mathbb {Z}^{i}$. Find three primes that have this property and the corresponding irreducibles.

Prove that $\mathbb {Z}^{\sqrt{-3}}$ is not a principal ideal domain.

In $\mathbb {Z}^{\sqrt{-5}}$, prove that $1+3\sqrt{-5}$ is irreducible but not prime.

In $\mathbb {Z}^{\sqrt{5}}$, prove that both $2$ and $1+\sqrt{5}$ are irreducible but not prime.

Prove that $\mathbb {Z}^{\sqrt{5}}$ is not a unique factorization domain.

Let $F$ be a field. Show that in $F[x]$ a non-zero prime ideal is a maximal ideal.

Trace through the argument given in Example 7 to find $q$ and $r$ in $\mathbb {Z}^{i}$ such that $3-4i = (2+5i)q+r$ and $d(r) < d(2+5i)$.

In $\mathbb {Z}^{\sqrt{2}} = \left\{ a+b\sqrt{2} \mid a, b \in \mathbb {Z}\right\}$, show that every element of the form $(3+2\sqrt{2})^{n}$ is a unit, where $n$ is a positive integer.

If $a$ and $b$ belong to $\mathbb {Z}^{\sqrt{d}}$, where $d$ is not divisible by the square of a prime and $ab$ is a unit, prove that $a$ and $b$ are units.

For a commutative ring with unity we may define associates, irreducibles, and primes exactly as we did for integral domains. With these definitions, show that both $2$ and $3$ are prime in $Z_{12}$ but $2$ is irreducible and $3$ is not.

Let $n$ be a positive integer and $p$ a prime that divides $n$. Prove that $p$ is prime in $Z_{n}$. (See Exercise 28.)

Let $p$ be a prime divisor of a positive integer $n$. Prove that $p$ is irreducible in $Z_{n}$ if and only if $p^{2}$ divides $n$. (See Exercise 28.)

Prove or disprove that if $D$ is a principal ideal domain, then $D[x]$ is a principal ideal domain.

Determine the units in $\mathbb {Z}^{i}$.

Let $p$ be a prime in an integral domain. If $p \mid a_{1}a_{2} \cdots a_{n}$, prove that $p$ divides some $a_{i}$. (This exercise is referred to in this chapter.)

Show that $3x^{2}+4x+3 \in Z_{5}[x]$ factors as $(3x+2)(x+4)$ and $(4x+1)(2x+3)$. Explain why this does not contradict the corollary of Theorem 18.3.

Corollary ($F[x]$ Is a UFD): Let $F$ be a field. Then $F[x]$ is a unique factorization domain.

Let $D$ be a principal ideal domain and $p$ an irreducible element of $D$. Prove that $D/ \left\langle p\right\rangle$ is a field.

Show that an integral domain with the property that every strictly decreasing chain of ideals $I_{1} \supset I_{2} \supset \cdots$ must be finite in length is a field.

An ideal $A$ of a commutative ring $R$ with unity is said to be finitely generated if there exist elements $a_{1}, a_{2}, \ldots , a_{n}$ of $A$ such that $A = \left\langle a_{1}, a_{2}, \ldots , a_{n}\right\rangle$. An integral domain $R$ is said to satisfy the ascending chain condition if every strictly increasing chain of ideals $I_{1} \subset I_{2} \subset \cdots$ must be finite in length. Show that an integral domain $R$ satisfies the ascending chain condition if and only if every ideal of $R$ is finitely generated.

Prove or disprove that a subdomain of a Euclidean domain is a Euclidean domain.

Show that for any nontrivial ideal $I$ of $\mathbb {Z}^{i}$, $\mathbb {Z}^{i}/I$ is finite.

Find the inverse of $1+\sqrt{2}$ in $\mathbb {Z}^{\sqrt{2}}$. What is the multiplicative order of $1+\sqrt{2}$?

In $\mathbb {Z}^{\sqrt{-7}}$, show that $N(6+2\sqrt{-7}) = N(1+3\sqrt{-7})$ but $6+2\sqrt{-7}$ and $1+3\sqrt{-7}$ are not associates.

Let $R = \mathbb {Z} \oplus \mathbb {Z} \oplus \cdots$ (the collection of all sequences of integers under componentwise addition and multiplication). Show that $R$ has ideals $I_{1}, I_{2}, I_{3}, \ldots$ with the property that $I_{1} \subset I_{2} \subset I_{3} \subset \cdots$. (Thus $R$ does not have the ascending chain condition.)

Prove that in a unique factorization domain, an element is irreducible if and only if it is prime.

Let $F$ be a field and let $R$ be the integral domain in $F[x]$ generated by $x^{2}$ and $x^{3}$. (That is, $R$ is contained in every integral domain in $F[x]$ that contains $x^{2}$ and $x^{3}$.) Show that $R$ is not a unique factorization domain.

Prove that for every field $F$, there are infinitely many irreducible elements in $F[x]$.

Prove that $\mathbb {Z}^{\sqrt{-2}}$ and $\mathbb {Z}^{\sqrt{2}}$ are unique factorization domains.

Express both $13$ and $5+i$ as products of irreducibles from $\mathbb {Z}^{i}$.

Find a mistake in the statement shown in Figure 18.2.

Let $I$ be a non-zero ideal in a principal ideal domain $R$. Prove that $R/I$ has a finite number of ideals.