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

Let \({\mathcal {N}}\) $\mathcal{N}$ be a 3-prime near-ring with center \(Z({\mathcal {N}})\) $Z(\mathcal{N})$ , \(\alpha , \beta : {\mathcal {N}}\rightarrow {\mathcal {N}}\) $\alpha ,\beta :\mathcal{N}\to \mathcal{N}$ be the maps, and J be a nonzero Jordan ideal of \({\mathcal {N}}.\) $\mathcal{N}.$ In this paper, we first introduce the notion of left \((\alpha , \beta )\) $(\alpha ,\beta )$ -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.