- Free module
In
mathematics , a free module is afree object in the category of modules. Given a set "S", a free module on "S" is a (particular construction of a) free module with basis "S".Every
vector space is free, and the free vector space on a set is a special case of a free module on a set.Definition
A free module is a module with a "free basis": a
linearly independent generating set.For an "R"-module "M", the set "E" = {"e"1, "e"2, ... "e""n"} is a free basis for "M" if:
# "E" is a generating set for "M", that is to say every element of "M" is a finite sum of elements of "E" multiplied by coefficients in "R";
# "E" is a free set, that is, if "r"1"e"1 + "r"2"e"2 + ... + "r""n""e""n" = 0, then "r"1 = "r"2 = ... = "r"n = "0" (where 0 is the zero element of "M" and "0" is the zero element of "R").If "R" has
invariant basis number , then by definition any two bases have the same cardinality. The cardinality of any (and therefore every) basis is called the rank of the free module "M", and "M" is said to be "free of rank n", or simply "free of finite rank" if the cardinality is finite.Note that an immediate
corollary of (2) is that the coefficients in (1) are unique for each "x".The definition of an infinite free basis is similar, except that "E" will have infinitely many elements. However the sum must be finite, and thus for any particular "x" only finitely many of the elements of "E" are involved.
In the case of an infinite basis, the rank of "M" is the
cardinality of "E".Construction
Given a set "E", we can construct a free "R"-module over "E". The module is simply the direct sum of "E" copies of "R", often denoted "R"("E"). We give a concrete realization of this direct sum, denoted by "C"("E"), as follows:
* As a set, "C"("E") contains the functions "f" : "E" → "R" such that "f"("x") = 0 for cofinitely many (all but finitely many) "x" in "E".
* Addition: for two elements "f", "g" ∈ "C"("E"), we define "f" + "g" ∈ "C"("E") by ("f" + "g")("x") = "f"("x") + "g"("x") for all "x" ∈ "E".
* Scalar multiplication: for α ∈ "R" and "f" ∈ "C"("E"), we define α"f" ∈ "C"("E") by (α"f")("x") = α"f"("x") for all "x" ∈ "E".A basis for "C"("E") is given by the set { δ"a" : "a" ∈ "E" } where:(a variant of the
Kronecker delta ).Define the mapping ι : "E" → "C"("E") by ι("a") = δ"a". This mapping gives a bijection between "E" and the basis vectors {δ"a"}"a"∈"E". We can thus identify these sets. Thus "E" may be considered as a linearly independent basis for "C"("E").
Universal property
The mapping ι : "E" → "C"("E") defined above is universal in the following sense. If φ is an arbitrary mapping from "E" to some "R"-module "M", then there exists a unique
module homomorphism ψ: "C"("E") → "M" such that φ = ψo ι.See also
*
Free object
Wikimedia Foundation. 2010.
