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 A note on $w$-Noetherian rings Bull. Korean Math. Soc. 2015 Vol. 52, No. 2, 541-548 https://doi.org/10.4134/BKMS.2015.52.2.541Published online March 31, 2015 Shiqi Xing and Fanggui Wang Sichuan Normal University, Sichuan Normal University 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. Keywords : $w$-moudle, $w$-finite type, $w$-Noetherian module, $w$-Noetherian ring MSC numbers : 13B02, 13A15, 13E99 Full-Text :