On Additive Mapping with Period 3 on Rings and Near-Rings

Abstract

In this research we introduced the definition of a map with period 3 on a ring Ɍ and on right ( left ) ideal Ῑ of Ɍ, then we prove that, when Ɍ is a prime ring with char (Ɍ)  2, and Ῑ  0, Ῑ is right ideal on Ɍ, if đ is a derivation with period 3 in Ɍ,then either đ=0, or u2=0 uῙ. Also we proved, when Ɍ is a domain with 1, and char (Ɍ)  6, If δ is a right generalized derivation on Ɍ with period 3, then δ is the identity map. Lastly, we define a map with Period 3 on near-rings, and gived results for prime left near-rings with maps acts as an anti-homomorphism (or homomorphism), with period 3, to obtain commuatative rings.