Centralizer and Jordan Centralizer of Inverse Semirings

Let M be a semiprime 2-torsion free inverse semiring, and let α be an endomorphism of M. Under some conditions, we prove a Jordan α-centralizer of M is a α-centralizer of M, also we prove if R: M→ M be an additive mapping such that R(r3) + α(r)R(r)α(r)' = 0 holds for all r ∈ M, where R is a centralizer, and α is a surjective endomorphism of M.