# free module

Let $R$ be a ring. A over $R$ is a direct sum of copies of $R$.

Similarly, as an abelian group is simply a module over $\mathbb{Z}$, a free abelian group is a direct sum of copies of $\mathbb{Z}$.

This is equivalent to saying that the module has a free basis, i.e. a set of elements with the property that every element of the module can be uniquely expressed as an linear combination over $R$ of elements of the free basis.

Every free module is also a projective module, as well as a flat module.

Title free module FreeModule1 2013-03-22 14:03:50 2013-03-22 14:03:50 Mathprof (13753) Mathprof (13753) 5 Mathprof (13753) Definition msc 16D40