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one should not confuse a schauder basis for Lp (which you can write down and is based on topological notions) with a hamel basis (which requires AOC and is based on algebraic notions). Tag: Reverted |
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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 those that do, [[free module]]s, need 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 spaces
=== Formal definition ===
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