On Weakly Primary Ideals

Weakly prime ideals in a commutative ring with non-zero identity have been introduced and studied in Anderson and Smith, Houston J. Math. 29: 831840, 2003. Here we study the weakly primary ideals of a commutative ring. We define a proper ideal 𝑃 of 𝑅 to be weakly primary if 0 𝑝𝑞 𝑃 implies 𝑝 𝑃 or 𝑞 Rad(𝑃), so every weakly prime ideal is weakly primary. Various properties of weakly primary ideals are considered. For example, we show that a weakly primary ideal 𝑃 that is not primary satisfies Rad(𝑃) Rad(0). Also, we show that an intersection of a family of weakly primary ideals that are not primary is weakly primary.