| Title:
|
Generalization of the $S$-Noetherian concept (English) |
| Author:
|
Dabbabi, Abdelamir |
| Author:
|
Benhissi, Ali |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
59 |
| Issue:
|
4 |
| Year:
|
2023 |
| Pages:
|
307-314 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
${\mathcal{S}}$-Noetherian |
| Keyword:
|
Nagata’s idealization |
| Keyword:
|
multiplicative system of ideals |
| MSC:
|
13A15 |
| MSC:
|
13B25 |
| MSC:
|
13E05 |
| idZBL:
|
Zbl 07790549 |
| idMR:
|
MR4641948 |
| DOI:
|
10.5817/AM2023-4-307 |
| . |
| Date available:
|
2023-08-15T13:30:42Z |
| Last updated:
|
2024-02-13 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/151788 |
| . |
| Reference:
|
[1] Anderson, D.D., Dumitrescu, T.: S-Noetherian rings.Comm. Algebra 30 (9) (2002), 4407–4416. MR 1936480, 10.1081/AGB-120013328 |
| Reference:
|
[2] Anderson, D.D., Winders, M.: Idealization of a module.J. Commut. Algebra 1 (2009), 3–53. MR 2462381, 10.1216/JCA-2009-1-1-3 |
| Reference:
|
[3] Hamann, E., Houston, E., Johnson, J.: Properties of uppers to zero in $R[X]$.Pacific J. Math. 135 (1988), 65–79. MR 0965685, 10.2140/pjm.1988.135.65 |
| Reference:
|
[4] Hamed, A., Hizem, S.: S-Noetherian rings of the form ${\mathcal{A}}[X]$ and ${\mathcal{A}}[[X]]$.Comm. Algebra 43 (2015), 3848–3856. MR 3360852 |
| Reference:
|
[5] Huckaba, J.A.: Commutative rings with zero divizors.Pure Appl. Math., Marcel Dekker, 1988. MR 0938741 |
| Reference:
|
[6] Lim, J.W., Oh, D.Y.: S-Noetherian properties on amalgamated algebras along an ideal.J. Pure Appl. Algebra 218 (2014), 1075–1080. MR 3153613, 10.1016/j.jpaa.2013.11.003 |
| . |
