Abbreviation: RRng

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: $\forall x\exists y(x\cdot y\cdot x=x)$

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 $h(-x)=-h(x)$.

$$\begin{examples} \end{examples}$$

$\begin{array}{lr} f(1)= &1 f(2)= & f(3)= & f(4)= & f(5)= & f(6)= & \end{array}$

Division rings

Rings with identity