### Table of Contents

## Rings with identity

Abbreviation: **Rng**$_1$

### Definition

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

$\langle R,+,-,0,\cdot\rangle $ is a ring

$1$ is an identity for $\cdot$: $x\cdot 1=x$, $1\cdot x=x$

##### Morphisms

Let $\mathbf{R}$ and $\mathbf{S}$ be 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 $\cdot$: $0\cdot x=0$ 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}$

Finite rings with identity in the Encyclopedia of Integer Sequences