### Table of Contents

## Regular rings

Abbreviation: **RRng**

### Definition

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

every element has a pseudo-inverse: $\forall x\exists y(x\cdot y\cdot x=x)$

##### Morphisms

Let $\mathbf{R}$ and $\mathbf{S}$ be regular rings. 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 $h(-x)=-h(x)$.

\begin{examples} \end{examples}

### Properties

### Finite members

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

f(2)= &

f(3)= &

f(4)= &

f(5)= &

f(6)= &

\end{array}$