On Weakly δ-Semiprimary Ideals of Commutative Rings (original) (raw)

Let R be a commutative ring with 1 = 0. We recall that a proper ideal I of R is called a semiprimary ideal of R if whenever a, b ∈ R and ab ∈ I, then a ∈ √ I or b ∈ √ I. We say I is a weakly semiprimary ideal of R if whenever a, b ∈ R and 0 = ab ∈ I, then a ∈ √ I or b ∈ √ I. In this paper, we introduce a new class of ideals that is closely related to the class of (weakly) semiprimary ideals. Let I(R) be the set of all ideals of R and let δ : I(R) → I(R) be a function. Then δ is called an expansion function of ideals of R if whenever L, I, J are ideals of R with J ⊆ I, then L ⊆ δ(L) and δ(J) ⊆ δ(I). Let δ be an expansion function of ideals of R. Then a proper ideal I of R (i.e., I = R) is called a (δ-semiprimary) weakly δ-semiprimary ideal of R if (ab ∈ I) 0 = ab ∈ I implies a ∈ δ(I) or b ∈ δ(I). For example, let δ : I(R) → I(R) such that δ(I) = √ I. Then δ is an expansion function of ideals of R and hence a proper ideal I of R is a (δ-semiprimary) weakly δsemiprimary ideal of R if and only if I is a (semiprimary) weakly semiprimary ideal of R. A number of results concerning weakly δ-semiprimary ideals and examples of weakly δ-semiprimary ideals are given. 1991 Mathematics Subject Classification. Primary 13A05, 13F05. Key words and phrases. semiprimary ideal, weakly semiprimary ideal, weakly prime ideal, weakly primary ideal, δ-primary ideal, δ-2-absorbing ideal.