Strongly primary ideals in rings with zero-divisors

Abstract

Let A be an integral domain with quotient field K. A. Badawi and E. Houston called a strongly primary ideal I of A if whenever x, y ∈ K and xy ∈ I, we have x ∈ I or y ∈ I for some n ≥ 1. In this note, we study the generalization of strongly primary ideal to the context of arbitrary commutative rings. We define a primary ideal P of A to be strongly primary if for each a, b ∈ A, we have aP ⊆ bA or bⁿA ⊆ aⁿP for some n ≥ 1.