On the commutativity of quotient near-rings under specific identities

Abstract

Purpose The purpose of this paper is to investigate the commutativity of quotient near-rings endowed with a left derivation and a multiplier satisfying specific differential algebraic identities, and to clarify the role of the 3-prime condition in this context. Design/methodology/approach The study considers a right near-ring N together with a 3-prime ideal P and a nonzero semigroup ideal I⊈P, and defines induced structures on the quotient near-ring N/P. Known results on derivations, left derivations and multipliers on near-rings are combined with several new lemmas to derive sufficient conditions that ensure N/P becomes a commutative ring. Findings The main results show, in particular, that if a non-P-trivial left derivation and a suitable multiplier on N satisfy certain differential identities on I, then the quotient near-ring N/P is necessarily commutative. The paper also establishes that, under the imposed hypotheses, there is no non-P-trivial left derivation on N, emphasizing the strength of the structural assumptions. Originality/value This work extends existing commutativity criteria for prime and semiprime near-rings by focusing on quotient near-rings equipped simultaneously with left derivations and multipliers that obey new differential identities. By restricting the analysis to suitable ideals of N, the paper highlights new mechanisms through which derivations and multipliers enforce commutativity in quotient near-ring structures.