# #3: Rings

The point of this post is to provide examples and non-examples for the following ring classes:

1. Ring
2. Commutative Ring
3. Integral Domain
4. Integrally Closed Domain
5. Unique Factorization Domain (UFD)
6. Principal Ideal Domain (PID)
7. Euclidean Domain
8. Field

This order forms an inclusion that goes from up to down. That is, the set of rings contains the set of commutative rings, the set of UFDs contains the set of PIDs, etc.

First we define a ring $R$ to be a non-empty set, together with two operations called addition and multiplication, denoted by $+$ and $\cdot$ respectively, which satisfy the following axioms:

• $R$ is an abelian group with respect to $+$ with identity element 0, i.e.:
• Closure under addition.
• Associativity under addition.
• Commutative under addition.
• Existence of additive identity 0.
• Existence of additive inverse $-x$ such that $-x + x = 0$ for all $x \in R$.
• Multiplication $\cdot$ is closed, associative, and has identity element 1 satisfying $1 \cdot x = x \cdot 1 = x$ for all $x \in R$. (However, it does not require inverse or commutativity. Beware that some definitions do not require multiplicative identity.)
• The left and right distributive laws hold, i.e.
• $a \cdot (b+c) = (a \cdot b) + (a \cdot c)$.
• $(a + b) \cdot c = (a \cdot c) + (b \cdot c)$.

Many familiar spaces are rings. In fact, by the inclusion chain, every object given below is a ring.

1. Rings $\supset$ Commutative Rings

A commutative ring is a ring where multiplication is commutative.

Example of a ring which is not a commutative ring:

• Matrix multiplication is not commutative. For example, $\begin{bmatrix} 0 & 1 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 1 \\ 0 & 0 \end{bmatrix} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$, while $\begin{bmatrix} 1 & 1 \\ 0 & 0 \end{bmatrix} \begin{bmatrix}0 & 1 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 0 & 2 \\ 0 & 0 \end{bmatrix}$.

2. Commutative Rings $\supset$ Integral Domains

An integral domain, also called a commutative domain, is a commutative ring that is also a domain. That is, it contains no zero-divisors. So if $a, b \in R$ and $a \neq 0, b\neq 0$, then $ab \neq 0$. Note that our definition is consistent as we require the existence of multiplicative identity in a ring.

Examples of commutative rings which are not integral domains:

• The quotient rings $\mathbb{Z}/n\mathbb{Z}$ where $n$ is a composite number. Let $n = ab$, where $a \neq 0,b \neq 0$. Then $ab = 0$.
• The continuous functions on $[0,1]$. Let $f(x)$ be $0$ on $[0,\frac{1}{2})$ and $-1 + 2x$ on $[\frac{1}{2},1]$, and let $g(x)$ be $1 - 2x$ on $[0,\frac{1}{2})$ and $0$ on $[\frac{1}{2},1]$. Then $f\cdot g = 0$ on $[0,1]$ even though neither $f$ nor $g$ is the zero function.

3. Integral Domains $\supset$ Integrally Closed Domains

An integrally closed domain is an integral domain that is integrally closed in its field of fractions.

Examples of integral domains which are not integrally closed domains:

• $\mathbb{Z}[\sqrt{-3}]$. For instance, the element $\frac{1+\sqrt{-3}}{2}$ is integral over $\mathbb{Z}[\sqrt{-3}]$ and is in the quotient field, but is not in $\mathbb{Z}[\sqrt{-3}]$. This is due to $\mathbb{Z}[\sqrt{-3}]$ being a finite index subgroup of $\mathbb{Z}[\frac{1+\sqrt{-3}}{2}]$. In particular, it has index 2. (On the other hand, $\mathbb{Z}[\sqrt{-5}]$ is integrally closed in its quotient field. Note that $\frac{1+\sqrt{-3}}{2}$ is a sixth root of unity and is thus algebraic, whereas $\frac{1+\sqrt{-5}}{2}$ is not algebraic.)
• $\mathbb{C}[x,y]/(x^2 - y^3)$, as it is a subring of the polynomial ring $\mathbb{C}[t]$ generated by $t^2$ and $t^3$. There is a singularity at $\mathbf{0}$.

4. Integrally Closed Domains $\supset$ Unique Factorization Domains

A unique factorization domain (UFD) is a commutative domain that satisfies:

• Every non-zero, non-unit of $R$ is a product of a finite number of irreducible elements.
• If $r$ is a non-zero, non-unit of $R$ with $r = u p_1 \cdots p_n = v q_1 \cdots q_m$ for some units $u, v$ and irreducibles $p_i, q_j$, then
• $n = m$,
• There exists a one-to-one and onto function $\sigma : \{1, \dots, n\} \to \{1, \dots, n\}$ and units $u_i$ such that $q_{\sigma(i)} = u_i p_i$ for $i \in \{1, \dots, n \}$.

Example of an integrally closed domain which is not a UFD:

• $\mathbb{Z}[\sqrt{-5}]$. For instance, the number $6$ can be factored the usual way as $(2)(3)$, or by the unusual way as $(1 + \sqrt{-5})(1 - \sqrt{-5})$. These are different factorizations as the only units in $\mathbb{Z}[\sqrt{-5}]$ are $1$ and $-1$.

5. Unique Factorization Domains $\supset$ Principal Ideal Domains

A principal ideal domain (PID) is a commutative domain where every ideal is principal.

Examples of UFDs which are not PIDs:

• $\mathbb{Z}[x]$, the polynomials with coefficients in $\mathbb{Z}$.
• $F[x_1, \dots, x_n]$, the polynomial ring in $n$ variables, if $n > 1$.

6. Principal Ideal Domains $\supset$ Euclidean Domains

A euclidean domain is a commutative domain such that there exists a function $d : R \setminus 0 \to \mathbb{Z}$ which takes on non-negative values and for every $a, b \in R$ with $b \neq 0$, there exist $q, r \in R$ such that $a = qb + r$ with either $r = 0$ or $d(r) < d(b)$.

Examples of PIDs which are not euclidean domains:

• The ring of integers of $\mathbb{Q}[\sqrt{d}]$ where $d = -19, -43, -67, -163, \dots$.

7. Euclidean Domains $\supset$ Fields

A field is a commutative ring that has a multiplicative inverse for every non-zero element.

Examples of euclidean domains which are not fields:

• The integers $\mathbb{Z}$. The only integers with inverses in $\mathbb{Z}$ are $1$ and $-1$.
• $F[x]$, the formal polynomials over a field $F$.
• $F[[x]]$, the formal power series over a field $F$.

Examples of fields:

• The finite fields $\mathbb{F}_q = \{0, 1, \dots, q-1\}$, with characteristic $q$ a prime number.
• $\mathbb{Q}[\sqrt{2}]$, the set of numbers in the form $a + b\sqrt{2}$, where $a$ and $b$ are rational.
• The rational numbers $\mathbb{Q}$.
• The real numbers $\mathbb{R}$.
• The complex numbers $\mathbb{C}$.

By the inclusion chain, fields also belong in every ring class given above.

For more information about certain types of rings, check out the following:

• Yep, formally constructing fields like $\mathbb{Z}$, $\mathbb{Q}$, and $\mathbb{R}$ for the first time is pretty mind-blowing. It’s like they just naturally arise from the axioms.