Jordan ideals and \((\alpha , \beta )\)-derivations on 3-prime near-rings and rings

Abstract

Let \({\mathcal {N}}\) be a 3-prime near-ring with center \(Z({\mathcal {N}})\), \(\alpha , \beta : {\mathcal {N}}\rightarrow {\mathcal {N}}\) be the maps, and J be a nonzero Jordan ideal of \({\mathcal {N}}.\) In this paper, we first introduce the notion of left \((\alpha , \beta )\)-derivations and then study their properties. We also characterize the commutativity of 3-prime near rings and obtain some related results. Finally, some examples are provided to illustrate that the hypotheses of our results are not superfluous.