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.
Abstract Algebra – Why Are Modules Called Modules?
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I think the following article:
Gregory H. Moore. The axiomatization of linear algebra: 1875-1940. Historia Mathematica, Volume 22, Issue 3, 1995, Pages 262–303
(Available here from Elsevier) may shed some light on your question, although you may not have enough mathematical experience to understand the entire article. Here is my understanding having browsed the article, but I must stress that I am not a mathematical historian, so please don't quote me!
The idea of an abstract space where an addition is defined between elements and there is a field action (rather than a particular realization as, for instance, $\mathbb{R}^n$ or $C([0,1])$) seems to be due to Peano in 1888, where he called them linear systems. The definition of an abstract vector space didn't catch on until the 1920s in the work of Banach, Hahn, and Wiener, each working separately. Hahn defined linear spaces in order to unify the theory of singular integrals and Schur's linear transformations of series (both employing infinite dimensional spaces). Wiener introduced vector systems which seems to be roughly equivalent to Banach's definition, which was motivated by finding a common framework to understand integral operators (Banach's 1922 paper "Sur les operations dans les ensembles abstraites et leur application aux équations intégrales" is available online and is quite readable) which were defined on champs (domains).
I understand the modern name vector space is popular because of a widely circulated 1941 textbook by Birkhoff and MacLane, A Survey of Modern Algebra, where the term is used.
As Asaf and Hans have indicated in their comments, the motivation for calling such spaces vector spaces is because intuitively, they generalize our understanding of "vectors" (differences between points) in a finite dimensional Euclidean. The motivation for calling such spaces linear spaces is because our ability to add together different elements is the crucial feature which lets us apply the general theory to solve specific problems which are not obviously (to the 1920's eye) about vectors (in particular, in PDE and mathematical physics).
In your course, it is unlikely you will cover material that requires this abstraction, but it is a good habit for later mathematics to work in generality while you maintain your intuition in concrete examples.
Well of course this has historic reasons. I don't know the details, though. But I would like to explain why the notion of an algebra over a ring, suitably generalized, is fundamental.
There are various notions which look very similar:
- ring
- monoid
- algebra over a ring
- normed algebra
- Banach algebra
- sheaf of rings
- topological monoid
- topological ring
- ring spectrum
- ...
Category theory is the unique field of mathematics where "similar" things are united to "one" thing. And in fact, in the context of monoidal categories, the mentioned examples are actually instances of one single notion: Monoid object, often also called "algebra object". One just has to apply this notion to different monoidal categories. In the above examples, these are:
- abelian groups
- sets
- modules over a ring
- normed vector spaces
- Banach spaces
- sheaves of abelian groups
- topological spaces
- topological abelian groups
- symmetric spectra
Best Answer
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.