Modules, Free Module, Basis

A Basis for a Module

A set of elements S in a module M is linearly independent if a finite linear combination of these elements totals to 0 only when the coefficients are all 0. As a corollary, each linear combination produces a unique element of M. If two different combinations produce x, subtract them to find a nontrivial linear combination that produces 0.

The independent set S is a basis if it spans (generates) all of M. Thus S is a basis iff each element in M is represented by a unique linear combination of members of S.

Free Module

A free R module M is a direct sum of copies of R. Elements of M are added and scaled per component. For now, we'll call this the definition of a "free module", but if you study category theory, you'll find that a free object has a more general definition, and when that definition is applied to the category of unitary modules, the "direct sum" definition falls out. But I digress.

Note that the free module M is both a left and a right module.

Given a free module M, project 1 onto each component, that is, 1 in each copy of R, and place these "ones" in a set S. It is easy to show that this is in fact a basis for M. Furthermore, R commutes with the basis elements, even if the members of R don't commute with each other.

A free module M has a basis, but how about the converse? Let M be a module with basis S. Let w be any element of S. Map the ring R into the module M via R*w. This is a ring homomorphism that maps onto the submodule generated by w. Since w is part of an independent set, the map is 1-1, hence a ring isomorphism. Now every element w in S generates a submodule of M that looks just like R. Since S is a basis, the members of M correspond 1-1 to linear combinations of the elements in S. Furthermore, addition and scaling are implemented by performing the same operations on basis coefficients. Thus M is isomorphic to the direct sum of copies of R, one copy for each basis element. In other words, M is a free module.

Modules over a Division Ring are Free

If R is a division ring, every module is a free module with a basis and a specific dimension or rank. This is discussed in detail in linear algebra.

Rank

If R is commutative, every free R module has a well defined rank. See the proof here.

Direct Sum

The direct sum of free modules is free. Take the union of the basis elements over each component to build a basis for M.

But the converse is not true. Z₆ is the direct product of Z₂ and Z₃, yet the two summands are not free Z₆ modules.

Matrices

Once a homomorphism is defined on the basis of M, it is defined on all of M. These module homomorphisms are called linear transformations in the world of linear algebra, and they can be represented by matrices, where matrix multiplication implements function composition. The ring of endomorphisms, described in the previous section, is actually the ring of n×n matrices over R. The module automorphisms correspond to the nonsingular matrices.