1-Absorbing δ-Primary Subhypermodules in Krasner Hypermodules

Let R be a commutative Krasner hyperring with 0≠1 and M a unital R-hypermodule. For an expansion δ on the lattice of subhypermodules of M, we introduce and study 1-absorbing δ-primary subhypermodules, defined by the condition that (ab)·m ∈ N for nonunits a, b ∈ R forces ab ∈ (N:M) or m ∈ δ(N). We establish equivalent colon and absorption characterizations, prove closure under directed unions in finitely generated hypermodules, obtain a correspondence with 1-absorbing δR-primary hyperideals in multiplication hypermodules under a quotient-preserving compatibility, and investigate behavior under surjective homomorphisms, quotients, and localization at prime hyperideals, where in the localization result we assume that δ is compatible with localization (and finite generation is used to ensure properness).