**A Survey of Modern Algebra**

**A Survey of Modern Algebra** === Chapter 1: The Integers --- HackMD **Book Mode** turns lists of links into a book . You could group links under header tags and create chapter-like sections. [Learn more here](https://hackmd.io/c/tutorials/%2Fs%2Fhow-to-create-book). Choose **Book Mode** in the top right sharing menu and hit "**Preview**" to see your book. #### 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. ::: ##### 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 $R$.