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Thanks to Conifold and Chris Leary's comments, I learned that the Nakayama lemma was not first created by a mathematician named Nakayama, but that mathematicians named Azumaya and Krull first created a lemma corresponding to the prototype of the Nakayama lemma. And I recently searched Azumaya algebra. However, the year in which this Azumaya algebra was created was 1951, which is the same as the Nakayama lemma. And since Nakayama lemma is used in the process of proving equivalent propositions within Azumaya algebra, I wondered if Azumaya algebra was behind the creation of Nakayama lemma.

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    $\begingroup$ "In spite of the name, Nakayama’s Lemma is due to Azumaya 51", Facchini, Module Theory. Azumaya references his joint paper with Nakayama for the concept of maximally central algebra. "This simple but important lemma is due to T. Nakayama, G. Azumaya and W. Krull. Priority is obscure, and although it is usually called the Lemma of Nakayama, late Prof. Nakayama did not like the name", Matsumura, Commutative Algebra. $\endgroup$
    – Conifold
    Commented Mar 7 at 12:45
  • $\begingroup$ @Conifold Hello, thank you for your comment. So, did T.Nakayama, G.Azumaya, W.Krull show no interest in Hilbert nullstellensatz and created their lemma? If yes, what was the motivation? $\endgroup$ Commented Mar 7 at 14:12
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    $\begingroup$ In his book Commutative Rings, Kaplansky references pp 212-3 of Nagata's Local Rings for a history of Nakayama's Lemma. You might find something there. $\endgroup$ Commented Mar 9 at 17:26

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