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The title of my Q. says it all:

QUESTION:   Who defined and who coined: module?

Would it be Emmy Noether?

EDIT   In view of @anon's and KConrad's answers, and as it could have been expected, the situation is a bit complex. Thus it looks that while Dedekind coined module, the final notion was defined by Emmy Noether. By defined I mean that module at a minimum should be a generalization of linear spaces, abelian groups and ideals.

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I would say, in answer to your edit, that if that is your definition of "define" then Dedekind both coined and defined the term (just like he coined the term "ideal", as opposed to "ideal number"). Noether generalized, just like she generalized a lot of the theory of rings and ideals, but according to Stillwell in the work quoted by anon, she was fond of saying "Es steht schon bei Dedekind" ("It is already in Dedekind") when talking about ideals/modules. – Arturo Magidin Aug 23 '14 at 2:08
up vote 8 down vote accepted


MODULE. A JSTOR search found the English term in E. T. Bell’s “Successive Generalizations in the Theory of Numbers,” American Mathematical Monthly, 34, (1927), 55-75. Bell was describing the work of Dedekind, basing his account on Dedekind’s French article, “Sur la Théorie des Nombres entiers algébriques” (1877) Gesammelte mathematische Werke 3 pp. 262-298. Dedekind used the French word module to translate his German term Modul. Stillwell writes in the Introduction to his English translation, Theory of Algebraic Integers (1996, p. 5), “Dedekind presumably chose the name ‘module’ because a module M is something for which ‘congruence modulo M’ is meaningful.” Curiously le module had once before been translated into English but then it went into English as the MODULUS of a complex number. [John Aldrich]

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This is right: it goes back to Dedekind's work, although he didn't have the concept as abstract as Emmy Noether did later on. For Dedekind, the concept was what we'd call a (finite free) $\mathbf Z$-module. Another reference mentioning this is in volume 2 of Ewald's "From Kant to Hilbert", where the relevant part can be seen at… – KConrad Aug 22 '14 at 23:39
That 1927 Monthly article is also the online OED's earlist citation for this sense of "module". – bof Aug 23 '14 at 0:20

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