Resumen de On Graded 1-Absorbing delta-Primary Ideals

Rashid Abu Dawwas, Anass Assarrar, Jebrel M Habeb, Najib Mahdou

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