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title: '**A Survey of Modern Algebra**'

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**A Survey of Modern Algebra**
===


Chapter 1: The Integers
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#### 1.1 Commutative Rings; Integral Domains


:::info
:bulb: **Defintion.** Let $R$ be a set of elements $a, b, c, ...$ for which the sum $a + b$ and the product *ab* of any two elements $a$ and $b$ (distinct or not) of $R$ are defined. Then $R$ is called a **commutative ring**  if the following postulates (1) - (8) hold:
:::



:::info
:bulb:
1.  *Closure*. If $a$ and $b$ are in $R$, then the sum $a + b$ and the product $ab$ are in $R$.
2. *Uniqueness*. If $a$ = $a'$ and $b$ = $b'$ in $R$, then $a + b = a' + b'$ and $ab = a'b'$. 
3. *Commutative laws*. For all $a$ and $b$ in $R$, $a + b = b + a$, $ab = ba$.
4.  
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##### Properties:
 

- (i) *Closure*. If $a$ and $b$ are in $R$, then the sum $a + b$ and the product $ab$ are in $R$.
- (ii) *Uniqueness*. If $a$ = $a'$ and $b$ = $b'$ in $R$, then $a + b = a' + b'$ and $ab = a'b'$. 
- (iii) *Commutative laws*. For all $a$ and $b$ in $R$, $a + b = b + a$, $ab = ba$.
- (iv) 


> Definition. An integral domain is a *commutative ring* in which the following additional postulate holds:

(ix) *Cancellation law*. If $c \neq 0$ and $ca = cb$, then $a = b$.


#### 1.2 Elementary Properties of Commutative Rings
Three basic laws for equality:
1. **Reflexive law**: $a=a$.
2. **Symmetric law**: If $a=b$, then $b=a$.
3. **Transitive law**: If $a=b$ and  $a=c$, then $a=c$ , valid for all $a$, $b$ and $c$.

***RULE 1***.  $(a+b)c = ac +bc$, for all $a$, $b$ and $c$ in $R$.




