01 - Rings

Ring

Ring is a datum

$$(R+,\times ,0,1)$$

where $R$ is a set and $1,0\in \mathbb{R}$ with binary operations $+,\times$ on $R$ such that

1. $R$ is an Abelian Group under $+$ with identity $+$

2. Binary operation $+$ is associative and $1\times x=x\times 1$ for all $x\in R$

3. Multiplication distributes over addition

$$\begin{array}{rl}x\times (y+z)& =(x\times y)+(x\times z)\\ (x+y)\times z& =(x\times z)+(y\times z)\forall x,y,z\in R\end{array}$$

For real numbers or integer, tend to use $\cdot$ instead of $\times$

Commutative Ring

Ring with property

$$x\cdot y=y\cdot x\forall x,y\in R$$

Examples of Rings

1. Integers of $\mathbb{Z}$
2. Modulo arithmetic $\mathbb{Z}/n\mathbb{Z}$ for some $n\in \mathbb{Z}$ (integers modulo $n$)
3. Gaussian Integrals

$$\mathbb{Z}[i]=\{a+ib:a,b\in \mathbb{Z}\}$$