Commutative graded-$n$-coherent and graded valuation rings

Year 2022, , 1047 - 1057, 01.08.2022

Anass Assarrar Najib Mahdou Ünsal Tekir Suat Koç

https://doi.org/10.15672/hujms.947574

Abstract

Let $R= \oplus_{ \alpha \in G} R_{\alpha}$ be a commutative ring with unity graded by an arbitrary grading commutative monoid $G$. For each positive integer, the notions of a graded-n-coherent module and a graded-n-coherent ring are introduced. In this paper many results are generalized from $n$-coherent rings to graded-$n$-coherent rings. In the last section, we provide necessary and sufficient conditions for the graded trivial extension ring to be a graded-valuation ring.

Keywords

coherent rings and modules, n-coherent rings and modules, graded modules and rings, graded-coherent rings and modules, graded-valuation rings, graded trivial extension ring

References

• [1] D.D. Anderson, D.F. Anderson and G.W. Chang, Graded-valuation domains, Comm. Algebra 45 (9), 4018-4029, 2017.
• [2] C. Bakkari, N. Mahdou and A. Riffi, Commutative graded-coherent rings, Indian J. Math. 61, 421-440, 2019
• [3] C. Bakkari, N. Mahdou and A. Riffi, Uniformly graded-coherent rings, Quaestiones Mathematicae, 44 (10), 1371-1391, 2021.
• [4] M. Bataineh and R. Abu-Dawwas, On graded 2-prime ideals, Mathematics, 9 (5), 493 (10 pages), 2021.
• [5] N. Bourbaki, Algèbre Commutative Chapitres 1-4, Masson, Paris, 1985.
• [6] N. Bourbaki, Algèbre, Chapitres 1-3, Springer-Verlag, Berlin, 2007.
• [7] G.W. Chang and D.Y. Oh, Discrete valuation overrings of a graded Noetherian domain, J. Commut. Algebra, 10 (1), 45-61, 2018.
• [8] D.L. Costa, Parameterizing families of non-Noetherian rings, Comm. Algebra, 22 (10), 3997-4011, 1994.
• [9] D. Dobbs, S.E. Kabbaj and N. Mahdou, n-Coherent rings and modules, Lecture Notes in Pure and Applied Mathematics, 269-282, 1996.
• [10] R. Gilmer, Commutative Semigroup Rings, Chicago, IL: University of Chicago Press, 1984.
• [11] S. Glaz, Commutative Coherent Rings, Lecture notes in mathematics 1371, Springer- Verlag, Berlin, 1989.
• [12] C. Nastasescu and F. Van Oystaeyen, Graded Ring Theory, North-Holland Math. Library, Amsterdam, 1982.
• [13] C. Nastasescu and F. Van Oystaeyen, Methods of Graded Rings, Lecture Notes in Math. 1836, Springer-Verlag, Berlin, 2004.
• [14] D.E. Rush, Noetherian properties in monoid rings, J. Pure Appl. Algebra, 185 (13), 259-278, 2003.
• [15] W.V. Vasconcelos, The rings of Dimension two. Marcel Dekker, New York, 1976.