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In this note, we give the Cohen type theorem for uniformly $S$-Noetherianmodules and the Eakin-Nagata type theorem for uniformly $S$-Noetherian rings.We also solve an open question proposed by Kim and Lim [5,Question 4.10].

A note on the Cohen type theorem and the Eakin-Nagata type theorem for uniformly $S$-Noetherian rings