# A NOTE ON w-NOETHERIAN RINGS

• Xing, Shiqi (College of Mathematics and Software Science Sichuan Normal University) ;
• Wang, Fanggui (College of Mathematics and Software Science Sichuan Normal University)
• Published : 2015.03.31

#### Abstract

Let R be a commutative ring. An R-module M is called a w-Noetherian module if every submodule of M is of w-finite type. R is called a w-Noetherian ring if R as an R-module is a w-Noetherian module. In this paper, we present an exact version of the Eakin-Nagata Theorem on w-Noetherian rings. To do this, we prove the Formanek Theorem for w-Noetherian rings. Further, we point out by an example that the condition (${\dag}$) in the Chung-Ha-Kim version of the Eakin-Nagata Theorem on SM domains is essential.

#### Acknowledgement

Supported by : National Natural Science Foundation of China

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