Jordan Ideals and Derivations Satisfying Algebraic Identities

Abstract

Let $$\mathcal {N}$$ be a 3-prime near-ring with center $$Z(\mathcal {N})$$ and $$\mathcal {J}$$ a nonzero Jordan ideal of $$\mathcal {N}$$ . The aim of this paper is to prove some theorems showing that $$\mathcal {N}$$ must be commutative if it admits a left multiplier F satisfying any one of the following properties: $$(i)\,F(\mathcal {J})\subseteq Z(\mathcal {N})$$ , $$(ii)\,F(\mathcal {J}^{2})\subseteq Z(\mathcal {N})$$ , $$(iii)\,F(ij)+[i, j]\in Z(\mathcal {N})$$ , $$(vi)\,F(ij)-ij+ji\in Z(\mathcal {N})$$ , $$(v)\,F(i\circ j)\in Z(\mathcal {N})$$ and $$(vi)\,F(i)G(j)\in Z(\mathcal {N}),$$ for all $$i, j\in \mathcal {J}.$$ Moreover, we give some examples which show that the hypotheses placed in our results are not superfluous.