A NOTE ON ω-NOETHERIAN RINGS
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Let R be a commutative ring. An R-module M is called a ω-Noetherian module if every submodule of M is of ω-finite type. R is called a ω-Noetherian ring if R as an R-module is a ω-Noetherian module. In this paper, we present an exact version of the Eakin-Nagata Theorem on ω-Noetherian rings. To do this, we prove the Formanek Theorem for ω-Noetherian rings. Further, we point out by an example that the condition (†) in the Chung-Ha-Kim version of the Eakin-Nagata Theorem on SM domains is essential.
Abstract
Let R be a commutative ring. An R-module M is called a ω-Noetherian module if every submodule of M is of ω-finite type. R is called a ω-Noetherian ring if R as an R-module is a ω-Noetherian module. In this paper, we present an exact version of the Eakin-Nagata Theorem on ω-Noetherian rings. To do this, we prove the Formanek Theorem for ω-Noetherian rings. Further, we point out by an example that the condition (†) in the Chung-Ha-Kim version of the Eakin-Nagata Theorem on SM domains is essential.
참고문헌 (15)
- D. D. Anderson, "Two star operations and their induced lattices", Comm. Algebra, 29(5), pp.2461~2475 (2000)
- G. W. Chang, "The w-integral closure of integral domains", J. Algebra, 295(1), pp.195~210 (2006)
- W. Chung, "Some remarks on strong Mori domains", Houston J. Math, 38(4), pp.1051~1059 (2012)
- P. M. Eakin, "The converse to a well known theorem on Noetherian rings", Math. Ann, 177, pp.278~282 (1968)
- E. Formanek, "Die st¨atze von Bertini f¨ur lokale Ringe", Math. Ann, 229, pp.97~111 (1977)
- H. Kim, "Injective modules over strong Mori domains", Houston J. Math, 34(2), pp.349~360 (2008)
- M. Nagata, "A type of subrings of a Noetherian ring", J. Math. Kyoto Univ, 8, pp.465~467 (1968)
- M. H. Park, "Groups rings and semigroup rings over strong Mori domains", J. Pure Appl. Algebra, 163(3), pp.301~318 (2001)
- M. H. Park, "On overrings of Strong Mori domains", J. Pure Appl. Algebra, 172(1), pp.79~85 (2002)
- F. G. Wang, "Finitely presented type modules and w-coherent rings", J. Sichuan Normal Univ, 33, pp.1~9 (2010)
- F. G. Wang, "On w-modules over strong Mori domains", Comm. Algebra, 25(4), pp.1285~1306 (1997)
- F. G. Wang, "On strong Mori domains", J. Pure Appl. Algebra, 135(2), pp.155~165 (1999)
- F. G. Wang, "Injective modules over w-Noetherian rings", Acta math. Sinica (Chin. Ser.), 53(6), pp.1119~1130 (2010)
- L. Xi, "On w-linked overrings", J. Math. Res. Exposition, 31, pp.337~346 (2011)
- Huayu Yin, "ω-MODULES OVER COMMUTATIVE RINGS", 대한수학회지, 48(1), pp.207~222 (2011), 대한수학회