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A Simple Proof of Cohen's Theorem

Let M be a module over a commutative ring R. Then M is called a Noetherian module if every submodule of M is finitely generated, and R is called a Noetherian ring if it is a Noetherian module over itself. Cohen proved that a commutative ring R is Noetherian if and only if every prime ideal in R is finitely generated (see, for example, [1] or [3]). Jothilingam has recently given a generalization of Cohen's theorem for