Free module

In mathematics, a free module is a module that has a basis, that is, a generating set consisting of linearly independent elements. Every vector space is a free module, but, if the ring of the coefficients is not a division ring (not a field in the commutative case), then there exist non-free modules. Given any set S and ring R, there is a free R-module with basis S, which is called the free module on S or module of formal R-linear combinations of the elements of S. A free abelian group is precisely a free module over the ring Z of integers. For a ring and an -module , the set is a basis for if: is a generating set for ; that is to say, every element of is a finite sum of elements of multiplied by coefficients in ; and is linearly independent if for every of distinct elements, implies that (where is the zero element of and is the zero element of ). A free module is a module with a basis. An immediate consequence of the second half of the definition is that the coefficients in the first half are unique for each element of M. If has invariant basis number, then by definition any two bases have the same cardinality. For example, nonzero commutative rings have invariant basis number. The cardinality of any (and therefore every) basis is called the rank of the free module . If this cardinality is finite, the free module is said to be free of finite rank, or free of rank n if the rank is known to be n. 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. Over a principal ideal domain (e.g., ), a submodule of a free module is free. If R is commutative, the polynomial ring in indeterminate X is a free module with a possible basis 1, X, X2, .... Let be a polynomial ring over a commutative ring A, f a monic polynomial of degree d there, and the image of t in B.