Shahabaddin Ebrahimi Atani, Farkhondeh Farzalipour

On Weakly Primary Ideals

Abstract:
Weakly prime ideals in a commutative ring with non-zero identity have been introduced and studied in [1]. Here we study the weakly primary ideals of a commutative ring. We define a proper ideal $P$ of $R$ to be weakly primary if $0 \neq p q \in P$ implies $p \in P$ or $q \in {\rm Rad} (P)$, 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 ${\rm Rad} (P) = {\rm Rad} (0)$. Also, we show that an intersection of a family of weakly primary ideals that are not primary is weakly primary.

Keywords:
Weakly primary, weakly prime, radical.

MSC 2000: 13C05, 13C13, 13A15