Editing Free module (section)

== Examples ==
Let ''R'' be a ring.
* ''R'' is a free module of rank one over itself (either as a left or right module); any unit element is a basis.
* More generally, If ''R'' is commutative, a nonzero ideal ''I'' of ''R'' is free if and only if it is a [[principal ideal]] generated by a [[nonzerodivisor]], with a generator being a basis.<ref>Proof: Suppose <math>I</math> is free with a basis <math>\{ x_j | j\}</math>. For <math>j \ne k</math>, <math>x_j x_k</math> must have the unique linear combination in terms of <math>x_j</math> and <math>x_k</math>, which is not true. Thus, since <math>I \ne 0</math>, there is only one basis element which must be a nonzerodivisor. The converse is clear.<math>\square</math></ref><!-- How about the non-commutative case? we at least need a reference for the non-commutative case. -->
* Over a [[principal ideal domain]] (e.g., <math>\mathbb{Z}</math>), a submodule of a free module is free.
* If ''R'' is commutative, the polynomial ring <math>R[X]</math> in indeterminate ''X'' is a free module with a possible basis 1, ''X'', ''X''<sup>2</sup>, ....
* Let <math>A[t]</math> be a polynomial ring over a commutative ring ''A'', ''f'' a monic polynomial of degree ''d'' there, <math>B = A[t]/(f)</math> and <math>\xi</math> the image of ''t'' in ''B''. Then ''B'' contains ''A'' as a subring and is free as an ''A''-module with a basis <math>1, \xi, \dots, \xi^{d-1}</math>.
* For any non-negative integer ''n'', <math>R^n = R \times \cdots \times R</math>, the [[Direct_product#Direct_product_of_modules|cartesian product]] of ''n'' copies of ''R'' as a left ''R''-module, is free. If ''R'' has [[invariant basis number]], then its [[rank of a module|rank]] is ''n''.
* A [[Direct sum of modules|direct sum]] of free modules is free, while an infinite cartesian product of free modules is generally ''not'' free (cf. the [[Baer–Specker group]]).
* A finitely generated module over a commutative [[local ring]] is free if and only if it is [[Flat module#Faithful flatness|faithfully flat]].<ref>{{harvnb|Matsumura|1986|loc=Theorem 7.10.}}</ref> Also, [[Kaplansky's theorem on projective modules|Kaplansky's theorem]] states a projective module over a (possibly non-commutative) local ring is free.
* Sometimes, whether a module is free or not is [[Undecidable_problem#Examples_of_undecidable_statements|undecidable]] in the set-theoretic sense. A famous example is the [[Whitehead problem]], which asks whether a Whitehead group is free or not. As it turns out, the problem is independent of ZFC.