We establish sufficient conditions for 3-prime near-rings to be commutative rings. In particular, for a 3-prime near-ring R with a derivation d, we investigate conditions such as \(d([U,V])\subseteq Z(R)\), \(d(U)\subseteq Z(R)\), \(x_{o}d(R)\subseteq Z(R)\), and \(Ux_{o}\subseteq Z(R)\). As a by-product, we generalize and extend known results related to rings and near-rings. Furthermore, we discuss the converse of a well-known result in rings and near-rings, namely: if \(x\in Z(R)\), then \(d(x)\in Z(R)\). In addition, we provide useful examples illustrating our results.

我们为三素近环成为可交换环建立了充分的条件。特别是，对于具有导数d的3素数近环R，我们研究诸如\（d（[U，V]）\ subseteq Z（R）\），\（d（U）\ subseteq Z （R）\），\（x_ {o} d（R）\ subseteq Z（R）\）和\（Ux_ {o} \ subseteq Z（R）\）。作为副产品，我们概括并扩展了与环和近环有关的已知结果。此外，我们讨论了在环和近环中众所周知的结果的反面，即：如果\（x \ in Z（R）\），则\（d（x）\ in Z（R）\）。此外，我们提供了有用的示例来说明我们的结果。