Abbreviation: SRng$_{01}$

A \emph{semiring with identity and zero} is a structure $\mathbf{S}=\langle S,+,0,\cdot,1 \rangle $ of type $\langle 2,0,2,0\rangle $ such that

$\langle S,+,0\rangle $ is a commutative monoids

$\langle S,\cdot,1\rangle$ is a monoids

$0$ is a zero for $\cdot$: $0\cdot x=0$, $x\cdot 0=0$

$\cdot$ distributes over $+$: $x\cdot(y+z)=x\cdot y+x\cdot z$, $(y+z)\cdot x=y\cdot x+z\cdot x$

Let $\mathbf{S}$ and $\mathbf{T}$ be semirings with identity and zero. A morphism from $\mathbf{S}$ to $\mathbf{T}$ is a function $h:S\rightarrow T$ that is a homomorphism:

$h(x+y)=h(x)+h(y)$, $h(x\cdot y)=h(x)\cdot h(y)$, $h(0)=0$, $h(1)=1$

Example 1:

$\begin{array}{lr} f(1)= & 1
f(2)= & 2
f(3)= & 6
f(4)= & 40
f(5)= & 295
f(6)= &3246
\end{array}$

Idempotent semirings with identity and zero

Rings with identity

Semirings with zero

Semirings with identity