On e-primary ideals of commutative rings

This work studies a novel generalization of primary ideals called ᶒ-primary ideals in commutative rings. We aim to investigate properties of primary ideals that extend naturally to ᶒ-primary ideals. If п is the proper ideals for the commutative rings with unity H, and ᶒ ∈ H an idempotent element such that ᶒ ∉ п. The ideals п is named ᶒ-primary ideals in the rings H if for h, k ∈ H with hk ∈ п, we have that ᶒh ∈ п or ᶒk ∈ rad(п). We settle and generalize multiple results from primary ideals to ᶒ-primary ideals. Furthermore, we study certain characterizations of ᶒ-primary ideals in the idealizations for the module H(+)M and amalgamations duplications for ring H ⋈Ψ z.