05 - Rings

Ring

A non-empty set $R$ with two binary operations $+$ and $\times$ is a ring if

1. $(R,+,0)$ is an abelian group
2. Multiplication $\times$ is associative
3. Distribution laws hold so for all $a,b,c\in R$ then

$$(a+b)c=ac+bc\text{ and }a(b+c)=ab+ac$$

Note that it DOES NOT NEED a multiplicative identity

Commutative Ring

Let $R$ be a ring then
For all $a,b\in R$ then

$$ab=ba$$

Note that any field is a commutative ring

Ring Homomorphism

Let $R,S$ be two rings

Map $\varphi :R\to S$ is a ring homomorphism if
For all $r,r^{\prime }\in R$

$$\varphi (r+r^{\prime })=\varphi (r)+\varphi (r^{\prime })\text{ and }\varphi (r{r}^{\prime })=\varphi (r)\varphi (r^{\prime })$$

Ring Isomorphism

Bijective Ring Homomorphism

Ideal

Non-empty subset $I$ of ring $R$ is an ideal if
For all $s,t\in I$ and $r\in R$ then

$$s-t\in I\text{ and }sr,rs\in I$$

Analogy between Ideals and Rings

Ideas are to rings as normal subgroups are to groups
In the sense that the set of additive cosets $R\setminus I$ inherit a ring structure from $R$ if $I$ is an ideal

For $r,r^{\prime }\in R$ then define

$$(r+I)+(r^{\prime }+I):=(r+r^{\prime })+I\text{and}(r+I)(r^{\prime }+I):=r{r}^{\prime }+I$$