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I know that a module is a generalization of a vector space, but I would like to know why are modules called modules?

Thanks for your kindly help.

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The name “module” was introduced by Dedekind in his work on ideals and number fields. You can find both his paper (in translation) *and* his explanation in Stillwell’s translation of Dedekind’s third exposition of the theory of ideals, “*Theory of Algebraic Integers*“, Cambridge University Press, Cambridge, 1996. It’s a very nice read, and it is (perhaps) suprisingly modern. Except for the fact that it takes what today would be considered an analytical detour, you could use this as a textbook in a class in algebraic number theory with almost no change in nomenclature, notation, or arguments.

Essentially, when working in number fields (finite extensions of $\mathbb{Q}$), and more specifically, in rings of integers of number fields (the collection of all elements in a finite extension $K$ of $\mathbb{Q}$ that satisfy a monic polynomial with integer coefficients), Dedekind isolated the necessary properties to be able to make “modular arguments”: closure under differences and absorption of multiplication. The idea was to reify Kummer’s notion of “ideal number”. Instead of inventing a new, not-really-existing-number to rescue unique factorization, you consider the collection of all elements that “would be” multiples of that ideal number. (It’s the same idea he used to defined real numbers as Dedekind cuts; instead of defining the numbers into existence, you identify a real number with the set of all rationals that “would be” less than or equal to the real number.)

Dedekind notes that the “collection of all multiples of $\alpha$” satisfies the conditions of being nonempty, closed under differences, and absorption of multiplication, and so this allows you to use “modular arguments” (as in, $a\equiv b\pmod{\alpha}$). So he called them modules, because you could “mod out” by them and do modular arithmetic.

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