Generalization of the $S$-Noetherian concept

Abdelamir Dabbabi and Ali Benhissi

Address: Mathematics Department, Faculty of Sciences of Monastir, Tunisia

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Abstract: Let $A$ be a commutative ring and ${\mathcal{S}}$ a multiplicative system of ideals. We say that $A$ is ${\mathcal{S}}$-Noetherian, if for each ideal $Q$ of $A$, there exist $I\in {\mathcal{S}}$ and a finitely generated ideal $F\subseteq Q$ such that $IQ\subseteq F$. In this paper, we study the transfer of this property to the polynomial ring and Nagata’s idealization.

AMSclassification: primary 13B25; secondary 13E05, 13A15.

Keywords: {\mathcal{S}}-Noetherian, Nagata’s idealization, multiplicative system of ideals.

DOI: 10.5817/AM2023-4-307