Last updates: 08 March 2011

$R/M$** is a field **$\iff M$ **is a maximal ideal.**

**Definition.**

- A
**Field**is a commutative ring $F$ such that if $x\in F$ and $x\ne 0$ 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 $\left(0\right)$ and not the whole ring $R$. - A
**maximal ideal**is an ideal $M$ of a ring $R$ such that- $M\ne R$,
- If $M\text{'}$ is an ideal of $R$ and $M\subseteq M\text{'}\ne R$ then $M=M\text{'}$.

Let $F$ be a commutative ring. Then $F$ is a field if and only if the only ideals onf $F$ are $\left(0\right)$ 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 **$\iff P$** is a prime ideal.**

**Definition.**

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

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

Let $R$ be an integral domain. Let ${F}_{R}=\left\{\frac{a}{b}\mid a,b\in R,b\ne 0\right\}$ be the set of fractions. Define fractions $a/b$, $c/d$ to be equal if $ad=bc$, i.e.,

$$a/b=c/d\phantom{\rule{2em}{0ex}}\text{if}\phantom{\rule{2em}{0ex}}ad=bc.$$ | 1.1 |

Let $R$ be an integral domain. Let ${F}_{R}=\left\{\frac{a}{b}\mid a,b\in R,b\ne 0\right\}$ be its set of fractions. Let equality of fractions be as defined in (1.1). Then the operations $+:{F}_{R}\hspace{0.17em}\times \hspace{0.17em}{F}_{R}\to {F}_{R}$ and $\times :{F}_{R}\hspace{0.17em}\times \hspace{0.17em}{F}_{R}\to {F}_{R}$ defined by

$$\frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd}\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}\frac{a}{b}\cdot \frac{c}{d}=\frac{ac}{bd}$$ | 1.2 |

Let $R$ be an integral domain and ${F}_{R}=\left\{\frac{a}{b}\mid a\in R,b\in R\setminus \left\{0\right\}\right\}$ be the set of fractions. Let equality of fractions be as defined in (1.1) and let operations $+:{F}_{R}\hspace{0.17em}\times \hspace{0.17em}{F}_{R}\to {F}_{R}$ and $\times :{F}_{R}\hspace{0.17em}\times \hspace{0.17em}{F}_{R}\to {F}_{R}$ 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 ${F}_{R}=\left\{\frac{a}{b}\mid a\in R,b\in R\setminus \left\{0\right\}\right\}$ with**equality of fractions**defined by $$\frac{m}{n}=\frac{p}{q}\phantom{\rule{2em}{0ex}}\text{if}\phantom{\rule{2em}{0ex}}mq=np$$ and operations of**addition**$+:{F}_{R}\hspace{0.17em}\times \hspace{0.17em}{F}_{R}\to {F}_{R}$ and**multiplication**$\times :{F}_{R}\hspace{0.17em}\times \hspace{0.17em}{F}_{R}\to {F}_{R}$ defined by $$\frac{m}{n}+\frac{p}{q}=\frac{mq+np}{nq}\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}\frac{m}{n}\cdot \frac{p}{q}=\frac{mp}{nq}.$$

**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 $\varphi :R\to {F}_{R}$ given by $$\begin{array}{cccc}\varphi :& R& \to & {F}_{R}\\ & r& \to & \frac{r}{1}\end{array}$$ is an injective ring homomorphism.

[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)