Letbe a near ring. An additive mapping : → is said to be a generalized semiderivation on if there exists a semiderivation :→ associated with a function : → such that ( ) = ( ) + ( ) ( ) = ( ) ( ) + ( ) and ( ( )) = ( ( )) for all , ∈ . In this paper we prove that prime near rings satisfying identities involving semiderivations are commutative rings, thereby extending some known results on derivations, semiderivations, and generalized derivations. We also prove that there exist no nontrivial generalized semiderivations which act as a homomorphism or as an antihomomorphism on a 3-prime near ring .
In this paper we investigate 3-prime near-rings with generalized two sided α-derivations satisfying certain differential identities. Consequently, some well known results have been generalized. Moreover, an example proving the necessity of the 3-primeness hypothesis is given.
The purpose of this paper is to investigate two sided α-derivations satisfying certain differential identities on 3-prime near-rings. Some well-known results characterizing commutativity of 3-prime near-rings by derivations (semi-derivations) have been generalized. Furthermore, examples proving the necessity of the 3-primeness hypothesis are given. 2010 AMS Mathematics subject classification. Primary 16N60, 16W25, 16Y30. Keywords and phrases. 3-prime near-rings, two sided α-derivations, commutativity.
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