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Was the term "module" in module over a ring originally chosen because of a relation to modular arithmetic?

Secondly, what is the etymology of the term "modular" in modular arithmetic? Does it have anything to do with "modular" in the sense of "consisting of multiple parts" (i.e. multiple modules), or something like that?

user56834
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  • Maybe better suited for https://hsm.stackexchange.com/. – Torsten Schoeneberg May 12 '18 at 18:13
  • What I'm quite sure of is that everything comes from Latin "modulo" ("in the manner of" or "to the base of" or very literally "in the little mode of"), in turn derived from modulus, in turn derived from modus. – Torsten Schoeneberg May 12 '18 at 18:16
  • @Torsten Schoeneberg: Absolutely sure, as a lot of other words. In France, it is avoiding as much as possible any double use of words (I believe that this should be the case in other countries). However sometimes it is inevitable. For example the word "$\bf{module}$" generalizes the notion of vector space but it is also used to designate the square root of the product of a complex $z$, that is, $| z |$. For congruences "$\bf{modulo}$" is used. – Piquito May 12 '18 at 18:28

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Jeff Miller's Earliest Known Uses of Some of the Words of Mathematics describes the origins of the word module as follows (boldface mine):

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]

As for modular arithmetic, very little is said there about the etymology, it is only stated that

Modular arithmetic is [first] found in English in 1941 in Fund. Mathematics by D. Harkin in the heading Finite modular arithmetic. [OED]

The site seems to be down at the moment, one can still read it from the web archive.

Arnaud Mortier
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  • One should add that according to this source, modulo was introduced by Gauß in the Disquisitiones Arithmeticae. While it appears unknown who first used the exact term "modular arithmetic", it seems obvious to me that it derives from Gauß' work. – Torsten Schoeneberg May 12 '18 at 19:36