**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