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Keywords:
$S$-finite; graded-$S$-coherent module; graded-$S$-coherent ring
Summary:
Recently, motivated by Anderson, Dumitrescu's $S$-finiteness, D. Bennis, M. El Hajoui (2018) introduced the notion of $S$-coherent rings, which is the $S$-version of coherent rings. Let $R= \bigoplus _{\alpha \in G} R_{\alpha }$ be a commutative ring with unity graded by an arbitrary commutative monoid $G$, and $S$ a multiplicatively closed subset of nonzero homogeneous elements of $R$. We define $R$ to be graded-$S$-coherent ring if every finitely generated homogeneous ideal of $R$ is $S$-finitely presented. The purpose of this paper is to give the graded version of several results proved in D. Bennis, M. El Hajoui (2018). We show the nontriviality of our generalization by giving an example of a graded-$S$-coherent ring which is not $S$-coherent and as a special case of our study, we give the graded version of the Chase's characterization of $S$-coherent rings.
References:
[1] Anderson, D. D., Anderson, D. F., Chang, G. W.: Graded-valuation domains. Commun. Algebra 45 (2017), 4018-4029. DOI 10.1080/00927872.2016.1254784 | MR 3627646 | Zbl 1390.13002
[2] Anderson, D. F., Chang, G. W., Zafrullah, M.: Graded Prüfer domains. Commun. Algebra 46 (2018), 792-809. DOI 10.1080/00927872.2017.1327595 | MR 3764897 | Zbl 1397.13001
[3] Anderson, D. D., Dumitrescu, T.: $S$-Noetherian rings. Commun. Algebra 30 (2002), 4407-4416. DOI 10.1081/AGB-120013328 | MR 1936480 | Zbl 1060.13007
[4] Assarrar, A., Mahdou, N., Tekir, Ü., Koç, S.: On graded coherent-like properties in trivial ring extensions. Boll. Unione Mat. Ital. 15 (2022), 437-449. DOI 10.1007/s40574-021-00312-6 | MR 4461706 | Zbl 1497.13002
[5] Bakkari, C., Mahdou, N., Riffi, A.: Commutative graded-coherent rings. Indian J. Math. 61 (2019), 421-440. MR 3971513 | Zbl 1451.13003
[6] Bakkari, C., Mahdou, N., Riffi, A.: Uniformly graded-coherent rings. Quaest. Math. 44 (2021), 1371-1391. DOI 10.2989/16073606.2020.1799106 | MR 4345229 | Zbl 1484.13003
[7] Bennis, D., Hajoui, M. El: On $S$-coherence. J. Korean Math. Soc. 55 (2018), 1499-1512. DOI 10.4134/JKMS.j170797 | MR 3883493 | Zbl 1405.13035
[8] Bourbaki, N.: Éléments de mathématique. Algèbre. Chapitres 1 à 3. Springer, Berlin (2007), French. DOI 10.1007/978-3-540-33850-5 | MR 0274237 | Zbl 1111.00001
[9] Chang, G. W., Oh, D. Y.: Discrete valuation overrings of a graded Noetherian domain. J. Commut. Algebra 10 (2018), 45-61. DOI 10.1216/JCA-2018-10-1-45 | MR 3804846 | Zbl 1400.13007
[10] Chase, S. U.: Direct products of modules. Trans. Am. Math. Soc. 97 (1960), 457-473. DOI 10.1090/S0002-9947-1960-0120260-3 | MR 0120260 | Zbl 0100.26602
[11] Gilmer, R.: Commutative Semigroup Rings. Chicago Lectures in Mathematics. University of Chicago Press, Chicago (1984). MR 0741678 | Zbl 0566.20050
[12] Glaz, S.: Commutative Coherent Rings. Lecture Notes in Mathematics 1371. Springer, Berlin (1989). DOI 10.1007/BFb0084570 | MR 0999133 | Zbl 0745.13004
[13] Huckaba, J. A.: Commutative Rings with Zero Divisors. Monographs and Textbooks in Pure and Applied Mathematics 117. Marcel Dekker, New York (1988). MR 0938741 | Zbl 0637.13001
[14] Kim, D. K., Lim, J. W.: When are graded rings graded $S$-Noetherian rings. Mathematics 8 (2020), Article ID 1532, 11 pages. DOI 10.3390/math8091532
[15] Năstăsescu, C., Oystaeyen, F. Van: Methods of Graded Rings. Lecture Notes in Mathematics 1836. Springer, Berlin (2004). DOI 10.1007/b94904 | MR 2046303 | Zbl 1043.16017
[16] Rush, D. E.: Noetherian properties in monoid rings. J. Pure Appl. Algebra 185 (2003), 259-278. DOI 10.1016/S0022-4049(03)00103-8 | MR 2006430 | Zbl 1084.13007
[17] Soublin, J.-P.: Anneaux et modules cohérents. J. Algebra 15 (1970), 455-472 French. DOI 10.1016/0021-8693(70)90050-5 | MR 0260799 | Zbl 0198.35803
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