Differential identities and commutativity in quotient near-rings over 3-prime ideals

This paper investigates commutativity conditions for quotient near-rings defined modulo 3-prime ideals. Specifically, we study the interplay between left derivations and multipliers satisfying certain differential identities, and we establish several new criteria ensuring that the corresponding quotient structure is commutative. Our approach relies on structural lemmas concerning semigroup ideals and central elements in quotient near-rings. The main theorems demonstrate that, under suitable algebraic conditions, the presence of such derivations and multipliers necessarily enforces commutativity in the quotient. Furthermore, we provide an explicit example to show that the 3-primeness assumption is indispensable: if this condition is relaxed, the obtained quotient may fail to be commutative despite satisfying all the imposed differential identities.