Weakly prime ideals

Let R be a commutative ring with identity. We define a proper ideal P of R to be weakly prime if 0 ≠ ab ∈ P implies a ∈ P or b ∈ P. For example, every proper ideal of a quasilocal ring (R, M) with M2 = 0 is weakly prime. We show that a weakly prime ideal P that is not prime satisfies P2 = 0, in fact, P√0 = 0. A number of results concerning weakly prime ideals and examples of weakly prime ideals are given. We show that every proper (principal) ideal of R is a product of weakly prime ideals if and only if R is a finite direct product of Dedekind domains (π-domains) and SPIR's or (R, M) is a quasilocal ring with M2 = 0.