Abstract In this paper, we introduce and study the class of (3, 1)-absorbing primary ideals in commutative rings. We establish several characterizations and examine their relationship with classical 1-absorbing primary ideals. Multiple criteria are provided, including conditions under which the two notions coincide and situations where they differ. Further connections are explored with other absorption-type ideals, and transfer results are established for trivial extensions and amalgamated algebras. Special attention is devoted to their classification in pullback rings, where explicit descriptions of radicals enable a precise determination of when such ideals are (3, 1)-absorbing primary. The results significantly enrich the hierarchy of absorption-type ideals, offering new insights into the relationships among primary, 1-absorbing primary, and 2-absorbing primary ideals, while highlighting their asymmetric absorption properties.
Abouhalaka et al. (Thu,) studied this question.