Is "module" over a ring historically related to "modular" arithmetic? 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?
 A: 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.
