On \(\zeta\)- derivations of additively inverse semirings

In this paper, we characterize $\zeta$-derivations of additively inverse semirings. In the sequel, some results concerning these derivations with commutator identities are investigated and as a consequence, some conditions are proved on $\zeta$-derivations which ensure the commutativity of semirings. In addition, we also explain how a $\zeta$-derivation maps ${\mathscr{S}}_{A}$ into its center and it is established that, if ${\mathscr{D}}$ is a $\zeta$-derivation of a non-commutative prime additively inverse semiring ${\mathscr{S}}_{A}$ with $\zeta:{\mathscr{S}}_{A}\rightarrow {\mathscr{S}}_{A}$, an onto morphism such that $({x_{1}}^{2})^{\mathscr{D}}+(x_{1})^{\mathscr{D}}{x_{1}}^{\prime}=0 $, $\forall$ $x_{1}\in{\mathscr{S}}_{A}$, then ${\mathscr{D}}=0$.