Badawi, AymanCelikel, Ece Yetkin2022-11-282022-11-282019-04-30Badawi, A., & Celikel, E. Y. (2019). On 1-absorbing primary ideals of commutative rings. In Journal of Algebra and Its Applications (Vol. 19, Issue 06, p. 2050111). World Scientific Publishing. https://doi.org/10.1142/s021949882050111x1793-6829http://hdl.handle.net/11073/25073Let 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.en-USPrime idealPrimary ideal1-absorbing primary ideal2-absorbing primary ideal2-absorbing idealWeakly prime idealWeakly primary idealWeakly 2-absorbing primary idealWeakly semiprime idealn-absorbing idealArticle10.1142/s021949882050111x