Euclidean / Principal Ideal / Unique Factorization Domain

I took Abstract Algebra I/II in UW-Madison(MATH541) with professor Chenxi Wu and Tonghai Yang. This article is inspired by prof. Wu's lecture note, and many formats, content, and structure of this blog are influenced by and similar to it.

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Please read this introduction with a complete understanding of ring.

Euclidean Domains

Definition 1. [Euclidean domain, Euclidean function]

An integral domain $D$ is Euclidean if there is a map $v : D\setminus\{0\} \rightarrow \mathbb{N}$ such that for any $a, b \in D$ and $b \neq 0$, $\exists\;q, r\in D$ such that (1) $a=qb+r$, and (2) $r=0$ or $v(r) < v(b)$. $D$ is called Euclidean domain and $v$ is called Euclidean function.

Example 2. $\mathbb{Z}=\{a + bi: a, b\in \mathbb{Z}\} \subseteq \mathbb{C}$ is an Euclidean domain as $v(a + bi) = a^2 + b^2$ is a Euclidean function.

Theorem 3. Let $D$ be an Euclidean domain, then any ideal $I$ of $D$ can be written as $ra$ for some $a \in I$.

Definition 4. [Principal Ideal, Generating Set]

1. Let $D$ be a communicative ring with identity. An ideal $\{ra : r \in D\}$ is called a principal ideal, written as $(a)$, with $a$ called the generator;
2. More generally, let $I$ be the index set, $\{a_i : i \in I\} \subseteq D$, then the ideal consists all of the elements of the form $\sum_{i\in I} r_ia_i$, with $r_i \in \mathbb{N}$, and all but finitely $r_i = 0$. Then, $\sum_{i\in I} r_ia_i$ is called the ideal generated by $\{a_i\}$, written as $(a_i, i \in I)$, and $\{a_i\}$ is called its generating set;

Definition 5. [Principal Ideal Domain(PID), Greatest Common Divisor]

1. An integral domain with every ideal is principal is called principal ideal domain(PID).

Definition 6. [Multiple/Divide/Divisor, Greatest Common Divisor]

1. $a$ is said to be a multiple of $b$ iff there is another element $m \in R$ with $a = mb$. In this case, we say $b$ divide $a$, or $b$ is a divisor of $a$.
2. From Theorem 3, we have for any $a, b \in D$, $(a, b) = (c)$. $c$ is called the greatest common divisor of $a$ and $b$, denoted as $c = gcd (a, b)$.

Definition 7. [Coprime] Let $R$ be a communicative ring with identity 1. For two ideals $I$, $J$ of $R$, we called $I$ and $J$ are coprime if $I + J = (1) = R$. For $a, b \in R$, if $(a) + (b) = R$, we called $a$ and $b$ coprime;

Proposition 1. If $a, b$ are non-zero elements in the communicative ring $R$ such that the ideal generated by $a$ and $b$ (that is, $(a, b)$) is a principal ideal $(d)$, then $d$ is a greatest common divisor of $a$ and $b$.

Definition 8. [Unit, Prime Ideal]

Let $D$ be a communicative ring with identity.

1. $u \in D$ is a unit if it has multiplicative inverse.
2. An ideal $P \subset R$ is a prime ideal if and only if $\forall ab \in P$, at least one of $a$ and $b$ is also in $P$.

Proposition 9. If $(d) = (d')$, we have $d = ud'$ for some unit $u \in R$. This gives if $d$ and $d'$ are both g.c.d. of $a$ and $b$, we have $d = ud'$ for some unit $u$.

Theorem 10. Let $R$ be a Euclidean Domain and let $a$ and $b$ be nonzero elements of $R$. Let $r_n$ be the last non-zero remainder in the Euclidean Algorithm for $a$ and $b$, then we have

1. $d$ is a g.c.d. of $a$ and $b$;
2. $d$ can be written as an R-linear combination of $a$ and $b$. There are $x, y \in R$ such that $$ d=ax+by $$

Principal Ideal Domains (PID)

Definition 11. [Principal Ideal Domain (P.I.D.)]

1. An integral domain is Principal Ideal Domain if and only if every ideal is principal.

Some examples:

1. $\mathbb{Z}$ is a PID.
2. $\mathbb{Z}[\sqrt{-5}]$ is not PID. A counterexample is $(3, 1 + \sqrt{-5})$, which is a nonprincipal ideal.

Proposition 12. Every nonzero prime ideal in a Principal Ideal Domain is a maximal ideal.

Corollary 13. If $R$ is any communicative ring such that the polynomiall ring $R[x]$ is a PID or Euclidean Domain, then $R$ is a field.

Definition 14. [Dedekind-Hasse Norm]

1. We define a norm $N$ to be a Dedekind-Hasse norm if $N$ is a positive norm and for every non-zero $a, b \in R$ either $a$ is an element of the ideal $(b)$ or there exists $s, t \in R$ with $0 < N(sa - tb) < N(b)$.

Proposition 15. Integral domain $R$ is a PID if and only if $R$ has a Dedekind-Hass Norm.

Unique Factorization Domain (UFD)

In terms of the integers $\mathbb{Z}$, we can factorize any integer $x \in \mathbb{Z}$ into primes $x = p_1^{a_1} p_2^{a_2} \cdots p_n^{a_n}$. This property can be extended to a much larger class of rings, namely Unique Factorization Domain. To introduce the formal definition of UFD, let's introduce some concepts shortly.

Definition 16. [Irreducible/Reducible, Prime, Associate]

Let $R$ be an integral domain.

1. Let $r \in R$ be a non-unit element. We say $r$ is reducible if and only if there are two non-unit elements $a, b \in R$ such that $r = ab$. Otherwise, we say $r$ is irreducible if and only if $\forall a, b \in R$ such that $r = ab$, at least one of $a, b$ is a unit.
2. A non-zero element $p \in R$ is called prime if and only if $(p)$ is a prime ideal. That is, for any $p | ab$, either $p | a$ or $p | b$.
3. Two elements $a, b \in R$ are said to be associate if and only if there exists a unit $u \in R$ such that $a = ub$.

Proposition 17. In an integral domain, every prime element is irreducible.

Proposition 18. In a PID, a non-zero element is prime if and only if it is irreducible.

Definition 19. [Unique Factorization Domain (UFD)]

An integral domain $R$ is an Unique Factorization Domain (UFD) if every nonzero and nonunit element $r \in R$ has the following two properties:

1. $r$ can be written as a finite product of irreducibles $r = p_1 p_2 \cdots p_n$;
2. For any other factorization of $r$ into irreducibles $r = q_1 q_2 \cdots q_m$, we have $m = n$, and we can renumber the factors so that $p_i$ is associate to $q_i$ for $i = 1, 2, \ldots, n$.

Proposition 20. In a UFD, a nonzero element is prime if and only if it is irreducible.

Proposition 21. Let R be an UFD, $a, b \in R$, suppose

$$ a = up_1^{e_1}p_2^{e_2} \cdots p_n^{e_n}, \quad b = vp_1^{f_1}p_2^{f_2} \cdots p_n^{f_n} $$

are prime factorizations for $a$ and $b$, where $u$ and $v$ are units, and the primes $p_1, p_2, \ldots, p_n$ are distinct and $e_i, f_i \geq 0$. Then, we have

$$ d = p_1^{\min(e_1, f_1)} p_2^{\min(e_2, f_2)} \cdots p_n^{\min(e_n, f_n)} $$

is a GCD of $a$ and $b$.

Theorem 22. Every PID is a UFD. In particular, every Euclidean Domain is a UFD.

Corollary 23. (Fundamental Theorem of Arithmetic) The integer ring $\mathbb{Z}$ is a UFD.

Corollary 24. Let $R$ be a PID, then there exists a multiplicative Dedekind-Hasse norm on $R$.