Free module

# Free module

Discussion

Encyclopedia
In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a free module is a free object
Free object
In mathematics, the idea of a free object is one of the basic concepts of abstract algebra. It is a part of universal algebra, in the sense that it relates to all types of algebraic structure . It also has a formulation in terms of category theory, although this is in yet more abstract terms....

in a category of module
Module (mathematics)
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...

s. Given a set $S$, a free module on $S$ is a free module with basis $S$. Every vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

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
Module (mathematics)
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...

with a basis
Basis (linear algebra)
In linear algebra, a basis is a set of linearly independent vectors that, in a linear combination, can represent every vector in a given vector space or free module, or, more simply put, which define a "coordinate system"...

: a linearly independent generating set. For an $R$-module $M$, the set $E\subseteq M$ is a basis for $M$ if:NEWLINE
NEWLINE
1. $E$ 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 $R$;
2. NEWLINE
3. $E$ is linearly independent, that is, if $r_1 e_1 + r_2 e_2 + \cdots + r_n e_n = 0_M$ for $e_1, e_2, \ldots , e_n$ distinct elements of $E$, then $r_1 = r_2 = \cdots = r_n = 0_R$ (where $0_M$ is the zero element of $M$ and $0_R$ is the zero element of $R$).
NEWLINE If $R$ has invariant basis number
Invariant basis number
In mathematics, the invariant basis number property of a ring R is the property that all free modules over R are similarly well-behaved as vector spaces, with respect to the uniqueness of their ranks.-Definition:...

, 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
Corollary
A corollary is a statement that follows readily from a previous statement.In mathematics a corollary typically follows a theorem. The use of the term corollary, rather than proposition or theorem, is intrinsically subjective...

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 still 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
Direct sum of modules
In abstract algebra, the direct sum is a construction which combines several modules into a new, larger module. The result of the direct summation of modules is the "smallest general" module which contains the given modules as submodules...

of  copies of $R$, often denoted $R^\left\{\left(E\right)\right\}$. We give a concrete realization of this direct sum, denoted by $C\left(E\right)$, as follows:NEWLINE
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• Carrier: $C\left(E\right)$ contains the functions $f:E\to R$ such that $f\left(x\right)=0$ for cofinitely many
Cofinite
In mathematics, a cofinite subset of a set X is a subset A whose complement in X is a finite set. In other words, A contains all but finitely many elements of X...

(all but finitely many) $x\in E$.
• NEWLINE
• Addition: for two elements $f,g\in C\left(E\right)$, we define $f+g\in C\left(E\right)$ by $\left(f+g\right)\left(x\right) = f\left(x\right) + g\left(x\right), \forall x\in E$.
• NEWLINE
• Inverse: for $f\in C\left(E\right)$, we define $\left(-f\right)\in C\left(E\right)$ by $\left(-f\right)\left(x\right) = -\left(f\left(x\right)\right), \forall x\in E$.
• NEWLINE
• Scalar multiplication: for $\alpha\in R, f\in C\left(E\right)$, we define $\alpha f\in C\left(E\right)$ by $\left(\alpha f\right)\left(x\right) = \alpha \left(f\left(x\right)\right), \forall x\in E$.
NEWLINE A basis for $C\left(E\right)$ is given by the set $\\left\{\delta_a : a\in E\\right\}$ where$\delta_a\left(x\right) = \begin\left\{cases\right\} 1, \quad\mbox\left\{if \right\} x=a; \\ 0, \quad\mbox\left\{if \right\} x\neq a \end\left\{cases\right\}$ (a variant of the Kronecker delta and a particular case of the indicator function, for the set $\\left\{a\\right\}$). Define the mapping $\iota : E\to C\left(E\right)$ by $\iota\left(a\right) = \delta_a$. This mapping gives a bijection between $E$ and the basis vectors $\\left\{\delta_a\\right\}_\left\{a\in E\right\}$. We can thus identify these sets. Thus $E$ may be considered as a linearly independent basis for $C\left(E\right)$.

## Universal property

The mapping $\iota : E\to C\left(E\right)$ defined above is universal
Universal property
In various branches of mathematics, a useful construction is often viewed as the “most efficient solution” to a certain problem. The definition of a universal property uses the language of category theory to make this notion precise and to study it abstractly.This article gives a general treatment...

in the following sense. If there is an arbitrary $R$-module $M$ and an arbitrary mapping $\varphi : E\to M$, then there exists a unique module homomorphism $\psi : C\left(E\right)\to M$ such that $\varphi = \psi\circ\iota$.