Editing Module (mathematics) (section)

=== Motivation ===
In a vector space, the set of [[scalar (mathematics)|scalars]] is a [[field (mathematics)|field]] and acts on the vectors by scalar multiplication, subject to certain axioms such as the [[distributive law]].  In a module, the scalars need only be a [[ring (mathematics)|ring]], so the module concept represents a significant generalization.  In commutative algebra, both [[ideal (ring theory)|ideals]] and [[quotient ring]]s are modules, so that many arguments about ideals or quotient rings can be combined into a single argument about modules.  In non-commutative algebra, the distinction between left ideals, ideals, and modules becomes more pronounced, though some ring-theoretic conditions can be expressed either about left ideals or left modules.<!-- (semi)perfect rings for instance have a litany of "Foo is true for all left ideals iff foo is true for all finitely generated left ideals iff foo is true for all cyclic modules iff foo is true for all modules" -->

Much of the theory of modules consists of extending as many of the desirable properties of vector spaces as possible to the realm of modules over a "[[well-behaved]]" ring, such as a [[principal ideal domain]]. However, modules can be quite a bit more complicated than vector spaces; for instance, not all modules have a [[basis (linear algebra)|basis]], and, even for those that do ([[free module]]s), the number of elements in a basis need not be the same for all bases (that is to say that they may not have a unique [[Free_module#Definition|rank]]) if the underlying ring does not satisfy the [[invariant basis number]] condition, unlike vector spaces, which always have a (possibly infinite) basis whose [[cardinality]] is then unique. (These last two assertions require the [[axiom of choice]] in general, but not in the case of [[finite-dimensional]] vector spaces, or certain well-behaved infinite-dimensional vector spaces such as [[Lp space|L<sup>''p''</sup> space]]s.)