Abbreviation: NRng$_1$
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}: $x\cdot 1=x\mbox{and}1\cdot x=x$
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 $h(-x)=-h(x)$.
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.
$0$ is a zero for $\cdot$: $0\cdot x=0$ and $x\cdot 0=0$.
$\begin{array}{lr}
f(1)= &1
f(2)= &
f(3)= &
f(4)= &
f(5)= &
f(6)= &
\end{array}$