## Near-fields

Abbreviation: **NFld**

### Definition

A ** near-field** is a near-rings with identity $\mathbf{N}=\langle N,+,-,0,\cdot,1
\rangle $ such that

$\mathbf{N}$ is non-trivial: $0\ne 1$

every non-zero element has a multiplicative inverse: $x\ne 0\Longrightarrow \exists y (x\cdot y=1)$

Remark: The inverse of $x$ is unique, and is usually denoted by $x^{-1}$.

##### Morphisms

Let $\mathbf{M}$ and $\mathbf{N}$ be near-fields. A morphism from $\mathbf{M}$ to $\mathbf{N}$ is a function $h:M\rightarrow N$ 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:

### Basic results

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

### Properties

### Finite members

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

### Subclasses

### Superclasses

### References

Trace: » near-fields