Let $(R,+,\cdot)$ be a finite semiring ($|R|<\infty)$ and let $0$ be the identity element of $(R,+)$.
Now the following property crossed my way: $$\forall a,b\in \left(R\setminus\{0\}\right):\;a+b\ne 0\;\;\text{and}\;\;a\cdot b\ne 0\;\;$$ This means nothing else but the identity element $0$ can not be generated by the other elements in $R$. In other words, there is no zero divisor and there is no element different from $0$ with an inverse element in $(R,+)$.
Now my question: This property seems not that special and so I wonder if there is a name for this property and maybe some results for the corresponding semirings?