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.
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机译：令$ R $为交换环。如果$ M $的每个子模块都是$ w $有限类型的，则$ R $模块$ M $称为$ w $ -Noetherian模块。如果$ R $作为$ R $模块是$ w $ -Noetherian模块，则$ R $称为$ w $ -Noetherian环。在本文中，我们给出了在$ w $ -Noetherian环上的Eakin-Nagata定理的精确版本。为此，我们证明了$ w $ -Noetherian环的Formanek定理。此外，我们通过一个例子指出，SM域上的Eakin-Nagata定理的Chung-Ha-Kim版本的条件（$ dag $）是必不可少的。
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