Communications of the
Korean Mathematical Society CKMS

We consider $\mathcal{N}$ to be a $3$-prime field and $\mathcal{P}$ to be a prime ideal of $\mathcal{N}.$ In this paper, we study the commutativity of the quotient near-ring $\mathcal{N}/\mathcal{P}$ with left multipliers and derivations satisfying certain identities on $P$, generalizing some well-known results in the literature. Furthermore, an example is given to illustrate the necessity of our hypotheses.

Keywords: Prime near-rings, derivations, left multipliers, commutativity

MSC numbers: Primary 16N60; Secondary 16W25, 16Y30