Bulletin of the
Korean Mathematical Society BKMS

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