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

## Fields, integral domains, fields of fractions

$R/M$ is a field $⇔M$ 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
1. $M\ne R$,
2. 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 $⇔P$ 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.

Definition.
• 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 $b\in R$, $b\ne 0$.

 $a/b=c/dif ad=bc.$ 1.1
Then equality of fractions is an equivalence relation on ${F}_{R}$.

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 $+:{F}_{R} × {F}_{R}\to {F}_{R}$ and $×:{F}_{R} × {F}_{R}\to {F}_{R}$ defined by

 $a b + c d = ad+bc bd and a b ⋅ c d = ac bd$ 1.2
are well defined.

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 $+:{F}_{R} × {F}_{R}\to {F}_{R}$ and $×:{F}_{R} × {F}_{R}\to {F}_{R}$ 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 $m n = p q if mq=np$ and operations of addition $+:{F}_{R} × {F}_{R}\to {F}_{R}$ and multiplication $×:{F}_{R} × {F}_{R}\to {F}_{R}$ defined by $m n + p q = mq+np nq and m n ⋅ p q = mp nq .$

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 $ϕ: R → FR r → r 1$ 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)