02. Basic Properties

2.1 Integral Domains

Zero Divisor

Let $R$ be a commutative ring then

Element $a\in R\setminus \{0\}$ is a zero divisor if

$$\exists b\in R\setminus \{0\}\text{ such that }a.b=0$$

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Integral Domain

Let $R$ be a ring then

$R$ is an Integral Domain if it is not the zero ring and has no zero divisors

Product is Zero Property $a,b\in R$ then

For

$$a.b=0⟹a=0\text{ or }b=0$$

Cancellation Property $x,y,z\in R$ then If

For

$$x.y=x.y$$

Then

$$x=0\text{ or }y=z$$

Proof

$$x.y=x.z⟺x.(y-z)=0$$

then result follows by definition

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Subrings of Integral Domains lemma

Let $R$ be an Integral Domain
Let $S$ be any subring of $R$ then

$$S\text{ is also an Integral Domain}$$

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Characteristic of Integral Domains

Let $R$ be a integral domain then

Characteristic of $R$

$$\mathrm{char}(R)=0\text{ or }p\text{ where prime }p\in Z$$

Proof

If $\mathrm{char}(R)=n>0$ then

$$\mathbb{Z}/n\mathbb{Z}\text{ is a subring of }R$$

So if $n=a.b$ where $a,b\in \mathbb{Z}$ where $a,b>1$ then

$$n_R=0⟺a_R.b_R=0$$

Hence $a_R$ and $b_R$ are both zero divisors as $a_R\ne 0$ and $b_R\ne 0$

So as $R$ is an integral domain then $\mathrm{char}(R)$ is either zero or prime

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Characteristic of Field

Characteristic of Field is always Zero or Prime

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Units of Ring

Let $R$ be a ring then subset

$$R^{\times }=\{r\in R:\exists s\in R\text{ such that }r.s=1\}$$

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Field of Fractions

Let $R$ be a ring then

$F(R)$ is the field of fractions of $R$

where ring $R$ embeds into $F(R)$ through map

$$r\mapsto \frac{r}{1}$$

with elements of $F(R)$ in form

$$\frac{a}{b}\text{ where }a,b\in R$$

and for $a,b,c,d\in R$

$$\frac{a}{b}=\frac{c}{d}⟺ad=bc$$

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Unique Injective Homomorphism

Let $k$ be a field
Let $\theta :R\to K$ be an embedding (injective homomorphism)

Then there exists unique injective homomorphism

$$\tilde{\theta }:F(R)\to K$$

that extends $\theta$ (${\tilde{\theta }}_R=\theta$ where $R$ is a subring of $F(R)$)

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