=====Near-rings with identity=====

Abbreviation: **NRng$_1$**
====Definition====
A \emph{near-ring with identity} is a structure $\mathbf{N}=\langle N,+,-,0,\cdot,1
\rangle $ of type $\langle 2,1,0,2,0\rangle $ such that


$\langle N,+,-,0,\cdot\rangle $ is a [[near-rings]]


$1$ is a \emph{multiplicative identity}:  $x\cdot 1=x\mbox{and}1\cdot x=x$

==Morphisms==
Let $\mathbf{M}$ and $\mathbf{N}$ be near-rings with identity. A morphism from $\mathbf{M}$
to $\mathbf{N}$ is a function $h:M\rightarrow N$ that is a homomorphism: 

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

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

====Examples====
Example 1: $\langle\mathbb{R}^{\mathbb{R}},+,-,0,\cdot,1\rangle$, the near-ring of functions on the real numbers with pointwise addition, subtraction, zero, composition, and the identity function.


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

====Properties====
^[[Classtype]]  |variety |
^[[Equational theory]]  |decidable |
^[[Quasiequational theory]]  | |
^[[First-order theory]]  | |
^[[Locally finite]]  |no |
^[[Residual size]]  |unbounded |
^[[Congruence distributive]]  |no |
^[[Congruence modular]]  |yes |
^[[Congruence n-permutable]]  |yes, $n=2$ |
^[[Congruence regular]]  |yes |
^[[Congruence uniform]]  |yes |
^[[Congruence extension property]]  | |
^[[Definable principal congruences]]  | |
^[[Equationally def. pr. cong.]]  | |
^[[Amalgamation property]]  | |
^[[Strong amalgamation property]]  | |
^[[Epimorphisms are surjective]]  | |
====Finite members====

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

====Subclasses====
[[Rings with identity]] 

====Superclasses====
[[Near-rings]] 


====References====

[(Ln19xx>
)]