On 1-absorbing primary ideals of commutative rings
Celikel, Ece Yetkin
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AbstractLet R be a commutative ring with nonzero identity. In this paper, we introduce the concept of 1-absorbing primary ideals in commutative rings. A proper ideal I of R is called a 1-absorbing primary ideal of R if whenever nonunit elements a,b,c ∈ R and abc ∈ I, then ab ∈ I or c ∈ √I. Some properties of 1-absorbing primary ideals are investigated. For example, we show that if R admits a 1-absorbing primary ideal that is not a primary ideal, then R is a quasilocal ring. We give an example of a 1-absorbing primary ideal of R that is not a primary ideal of R. We show that if a ring R is not a quasilocal, then a proper ideal I of R is a 1-absorbing primary ideal of R if and only if I is a primary ideal. We show that if R is a Noetherian domain, then R is a Dedekind domain if and only if every nonzero proper 1-absorbing primary ideal of R is of the form Pn for some nonzero prime ideal P of R and a positive integer n ≥ 1. We show that a proper ideal I of R is a 1-absorbing primary ideal of R if and only if whenever I1I2I3 ⊆ I for some proper ideals I1, I2, I3 of R, then I1I2 ⊆ I or I3 ⊆ √I.
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Badawi, Ayman; Celikel, Ece Yetkin (World Scientific, 2022)Let R be a commutative ring with 1 ≠ 0. In this paper, we introduce the concept of weakly 1-absorbing primary ideal which is a generalization of 1-absorbing primary ideal. A proper ideal I of R is called a weakly 1-absorbing ...