# Why are modules called modules?

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.

• See the entry here. This is a good place to look, in general. As one might expect, Dedekind and congruences pop up. – Dylan Moreland Feb 4 '12 at 15:53

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.