### Table of Contents

## Rings

Abbreviation: **Rng**

### Definition

A \emph{ring} is a structure $\mathbf{R}=\langle R,+,-,0,\cdot \rangle $ of type $\langle 2,1,0,2\rangle $ such that

$\langle R,+,-,0\rangle $ is an abelian groups

$\langle R,\cdot \rangle $ is a semigroups

$\cdot $ distributes over $+$: $x\cdot (y+z)=x\cdot y+x\cdot z$, $(y+z)\cdot x=y\cdot x+z\cdot x$

##### Morphisms

Let $\mathbf{R}$ and $\mathbf{S}$ be rings. A morphism from $\mathbf{R}$ to $\mathbf{S}$ is a function $h:R\to S$ that is a homomorphism:

$h(x+y)=h(x)+h(y)$, $h(x\cdot y)=h(x)\cdot h(y)$

Remark: It follows that $h(0)=0$ and $h(-x)=-h(x)$.

### Examples

Example 1: $\langle\mathbb{Z},+,-,0,\cdot\rangle$, the ring of integers with addition, subtraction, zero, and multiplication.

### Basic results

$0$ is a zero for $\cdot$: $0\cdot x=0$ and $x\cdot 0=0$.

### Properties

### Finite members

$\begin{array}{lr}
f(1)= &1

f(2)= &2

f(3)= &2

f(4)= &11

f(5)= &2

f(6)= &4

\end{array}$