We introduce and study sum–difference prime (sd-prime) ideals in noncommutative rings, extending the square–difference absorption property from the commutative setting. We establish fundamental relationships between sd-prime, prime, and semiprime ideals, and we analyze the stability of sd-primeness under standard ring constructions, obtaining structural classification theorems for sd-prime ideals in a wide range of related rings, including homomorphic images, quotients, matrix rings, direct products, trivial ring extensions, upper triangular matrix rings, and pullback rings. We further introduce noncommutative amalgamations, where we describe the ideal structure and obtain preservation results for sd-prime ideals, supported by explicit counterexamples. In the commutative setting, we develop an ideal-theoretic geometry for sd-prime ideals by introducing the sd–spectrum SdSpec(R) together with its sd-Zariski topology and the associated sd-radical, establishing spectral properties and identifying conditions under which closed sets admit unique generic points.
Abouhalaka et al. (Thu,) studied this question.