A module is a mathematical object in which things can be added together commutatively by multiplying coefficients and in which most of
the rules of manipulating vectors hold. A module is abstractly
very similar to a vector space, although in modules,
coefficients are taken in rings
that are much more general algebraic objects than the fields
used in vector spaces. A module taking its coefficients
in a ring 
is called a module over 
, or a R-module.
Modules are the basic tool of homological algebra. Examples of modules include the set of integers 
, the cubic lattice in 
dimensions 
, and the group ring of a group.
is a module over itself. It is closed
under addition and subtraction
(although it is sufficient to require closure under
subtraction). Numbers of
the form 
for 
and 
a fixed integer form a submodule since, for all 
,
 |
and 
is still in 
.
Given two integers 
and 
,
the smallest module containing 
and 
is the module for their greatest
common divisor, 
.
