On graded 1 -absorbing δ -primary ideals





graded 1-absorbing prime ideal, graded 1-absorbing δ -primary ideal, graded δ -primary ideal, trivial graded ring extension


Let G be an abelian group with identity 0 and let R be a commutative graded ring of type G with nonzero unity. Let I(R) be the set of all ideals of R and let δ: I(R)⟶I(R) be a function. Then, according to (R. Abu-Dawwas, M. Refai, Graded δ-Primary Structures, Bol. Soc. Paran. Mat., 40 (2022), 1-11), δ is called a graded ideal expansion of a graded ring R if it assigns to every graded ideal I of R another graded ideal δ(I) of R with I ⊆ δ(I), and if whenever I and J are graded ideals of R with J ⊆ I, we have δ (J) ⊆ δ(I). Let δ be a graded ideal expansion of a graded ring R. In this paper, we introduce and investigate a new class of graded ideals that is closely related to the class of graded δ-primary ideals. A proper graded ideal I of R is said to be a graded 1-absorbing δ-primary ideal if whenever nonunit homogeneous elements a,b,c ∊ R with abc ∊ I, then ab ∊  I or c ∊  δ(I). After giving some basic properties of this new class of graded ideals, we generalize a number of results about 1-absorbing δ-primary ideals into these new graded structure. Finally, we study the graded 1-absorbing δ-primary ideals of the localization of graded rings and of the trivial graded ring extensions.


R. Abu-Dawwas, On graded strongly 1-absorbing primary ideals. Khayyam Journal of Mathematics, Vol. 8, No. 1, pp. 42-52, 2022.

R. Abu-Dawwas, E. Yildiz, Ü. Tekϊr, and Koç, On graded 1-absorbing prime ideals. São Paulo Journal of Mathematical Sciences, Vol. 15, pp. 450-462, 2021.

R. Abu-Dawwas and M. Refai, Graded δ-Primary Structures. Bol. Soc. Paran. Mat., Vol. 40, pp. 1-11, 2022.

A. Assarrar, N. Mahdou, U. Tekir and S. Koç, On graded coherent-like properties in trivial ring extensions. Bolletino dell Unione Mat. Ital., Vol. 15, pp. 437-449, 2022.

K. Al-Zoubi, R. Abu-Dawwas and S. Ceken, On graded 2-absorbing and graded weakly 2-absorbing ideals. Hacettepe Journal of Mathematics and Statistics, Vol. 48, No. 3, pp. 724-731, 2019.

A. Badawi and Y. Celikel, On 1-absorbing primary ideals of a commutative rings. J. Algebra Appl., 2020, 2050111.

J. M. Habeb and R. Abu-Dawwas, Graded classical weakly prime submodules over non-commutative graded rings. submitted.

C. Nastasescu and F. Oystaeyen, Graded Ring Theory, Mathematical Library 28, North Holand, Amsterdam, 1982.

C. Nastasescu and F. Oystaeyen, Methods of graded rings, Lecture Notes in Mathematics, 1836, Springer-Verlag, Berlin, 2004.

R. N. Uregen, U. Tekir, K. P. Shum and S. Koç, On graded 2-absorbing quasi primary ideals. Southeast Asian Bull. Math., Vol. 43, No. 4, pp. 601-613, 2019.



How to Cite

R. Abu-Dawwas, A. Assarrar, J. M. Habeb, and N. Mahdou, “On graded 1 -absorbing δ -primary ideals”, Proyecciones (Antofagasta, On line), vol. 43, no. 3, pp. 571-586, May 2024.