§1T. Fields, Integral Domains, Fields of Fractions

§1T. Fields, Integals Domains, Fields of Fractions

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updates: 08 March 2011

Fields, integral domains, fields of fractions

$R / M$ is a field $\Leftrightarrow M$ is a maximal ideal.

Definition.

A Field is a commutative ring $F$ such that if $x \in F$ and $x \neq 0$ there exist an element $x^{- 1} \in F$ such that $x x^{- 1} = 1 .$
A proper ideal is an ideal of $R$ that is not the zero ideal $(0)$ and not the whole ring $R$ .
A maximal ideal is an ideal M of a ring R such that
1. $M \neq R$ ,
2. If $M'$ is an ideal of $R$ and $M \subseteq M' \neq R$ then $M = M'$ .

Let $F$ be a commutative ring. Then $F$ is a field if and only if the only ideals onf $F$ are $(0)$ and $F$ .

Let $R$ be a commutative ring and $M$ be an ideal of $R$ . Then

$R / M$ is a field if and only if $M$ is a maximal ideal.

$R / P$ is an integral domain $\Leftrightarrow P$ is a prime ideal.

Definition.

An integral domain is a commutative ring $R$ such that if $a, b \in R$ and $a b = 0$ then either $a = 0$ or $b = 0$ .
A zero divisor in a ring $R$ is an element $a \in R$ such that $a b = 0$ for some $b \in R$ , $b \neq 0$ .
A prime ideal is an ideal $P$ in a commutative ring $R$ such that if $a, b \in R$ and $a b \in P$ then either $a \in P$ or $b \in P$ .

HW: Show that an integral domain is a commutative ring with no zero divisors except $0$ .

(Cancellation Law) Let $R$ be an integral domain. If $a, b, c \in R$ , $c \neq 0$ , and $a c = b c$ , then $a = b$ .

Let $R$ be a commutative ring and let $P$ be an ideal of $R$ . Then

$R / P$ is an integral domain if and only if $P$ is a prime ideal.

Definition.

Let $R$ be an integral domain. A fraction is an expression $a / b$ , $a \in R$ , $b \in R$ , $b \neq 0$ .
A zero divisor in a ring $R$ is an element $a \in R$ such that $a b = 0$ for some $b \in R$ , $b \neq 0$ .

Let $R$ be an integral domain. Let $F_{R} = \{\frac{a}{b} ∣ a, b \in R, b \neq 0\}$ be the set of fractions. Define fractions $a / b$ , $c / d$ to be equal if $a d = b c$ , i.e.,

a / b = c / d if a d = b c .

1.1

Then equality of fractions is an equivalence relation on

F_{R}

Let $R$ be an integral domain. Let $F_{R} = \{\frac{a}{b} ∣ a, b \in R, b \neq 0\}$ be its set of fractions. Let equality of fractions be as defined in (1.1). Then the operations $+ : F_{R} \times F_{R} \to F_{R}$ and $\times : F_{R} \times F_{R} \to F_{R}$ defined by

\frac{a}{b} + \frac{c}{d} = \frac{a d + b c}{b d} and \frac{a}{b} \cdot \frac{c}{d} = \frac{a c}{b d}

1.2

are well defined.

Let $R$ be an integral domain and $F_{R} = \{\frac{a}{b} ∣ a \in R, b \in R ∖ \{0\}\}$ be the set of fractions. Let equality of fractions be as defined in (1.1) and let operations $+ : F_{R} \times F_{R} \to F_{R}$ and $\times : F_{R} \times F_{R} \to F_{R}$ be as given in (1.2). Then $F_{R}$ is a field.

Definition.

Let $R$ be an integral domain. The field of fractions of $R$ is a the set $F_{R} = \{\frac{a}{b} ∣ a \in R, b \in R ∖ \{0\}\}$ with equality of fractions defined by $\frac{m}{n} = \frac{p}{q} if m q = n p$ and operations of addition $+ : F_{R} \times F_{R} \to F_{R}$ and multiplication $\times : F_{R} \times F_{R} \to F_{R}$ defined by $\frac{m}{n} + \frac{p}{q} = \frac{m q + n p}{n q} and \frac{m}{n} \cdot \frac{p}{q} = \frac{m p}{n q} .$

HW: Give an example of an integral domain $R$ and its field of fractions.

Let $R$ be an integral domain with identity $1$ and let $F_{R}$ be its field of fractions. Then the map $ϕ : R \to F_{R}$ given by $\begin{matrix} ϕ : & R & \to & F_{R} \\ r & \to & \frac{r}{1} \end{matrix}$ is an injective ring homomorphism.

References

[CM] H. S. M. Coxeter and W. O. J. Moser, Generators and relations for discrete groups, Fourth edition. Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], 14. Springer-Verlag, Berlin-New York, 1980. MR0562913 (81a:20001)

[GW1] F. Goodman and H. Wenzl, The Temperly-Lieb algebra at roots of unity, Pacific J. Math. 161 (1993), 307-334. MR1242201 (95c:16020)

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