ring

Definition

A ring is a set $R$ that has two operations $+$ and $\cdot$. They must satisfy the ring axioms:

1. $R$ is an abelian group under addition
2. Multiplication is associative

$$(a\cdot b)\cdot c=a\cdot (b\cdot c)\forall a,b,c\in R$$

3. Multiplication is closed: $a\cdot b\in R$ for all $a,b\in R$.
4. Multiplication is distributive:

• left distributive $a\cdot (b+c)=a\cdot b+a\cdot c$
• right distributive $(a+b)\cdot c=a\cdot c+b\cdot c$

Morphisms

Morphisms in the category $\text{Ring}$ are called ring homomorphisms. A ring homomorphism is a map $f:R\to S$ which satisfies

1. $f(r_1+r_2)=f(r_1)+f(r_2)$
2. $f(r_1{r}_2)=f(r_1)f(r_2)$
3. $f(1_R)=1_S$

Relation to subrings and ideals

Let $f:R\to S$ be a ring homomorphism. Then

$$\text{im}f=\{s\in S|s=f(r)\text{for some }r\in R\}$$

is a subring of $S$. Similarly,

$$\ker f=\{r\in R|f(r)=0\}$$

is an ideal.

This is the main source of “finding” subring and ideals in practice.

Types of rings

Some important classes of rings are

• domain (ring theory)
• division ring
• field

Classes of rings

$$\text{field}⟹\text{division ring}⟹\text{domain}.$$