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

In this note, we give the Cohen type theorem for uniformly $S$-Noetherian modules 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].