### Table of Contents

## Commutative rings with identity

Abbreviation: **CRng$_1$**

### Definition

A \emph{commutative ring with identity} is a rings with identity $\mathbf{R}=\langle R,+,-,0,\cdot,1 \rangle$ such that $\cdot$ is commutative: $x\cdot y=y\cdot x$

##### Morphisms

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

$h(x+y)=h(x)+h(y)$, $h(x\cdot y)=h(x)\cdot h(y)$, $h(1)=1$

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

### Examples

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

### Basic results

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

### Properties

### Finite members

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

f(2)= &1

f(3)= &1

f(4)= &4

f(5)= &1

f(6)= &1

\end{array}$