Math 2.0

Associative rings and algebras Definition

Free module

Builds on Module over a ring

Definition

Let RR be a ring with identity element and let MM be a left module over RR. A subset E⊆ME \subseteq M is a basis of MM when:

  1. Each x∈Mx \in M is a finite sum r1e1+⋯+rkekr_1 e_1 + \cdots + r_k e_k with ri∈Rr_i \in R and ei∈Ee_i \in E.
  2. If e1,…,ek∈Ee_1, \dots, e_k \in E are distinct and r1e1+⋯+rkek=0r_1 e_1 + \cdots + r_k e_k = 0, then r1=⋯=rk=0r_1 = \cdots = r_k = 0.

The module MM is free when it has a basis.

Remarks

RnR^n is free, with the standard basis; each vector space is free over its field. For n≥2n \ge 2, the ℤ\Z-module ℤ/nℤ\Z / n\Z is not free, because nx=0n x = 0 for all its elements. Over a nonzero commutative ring, any two bases of a free module have the same cardinality, called the rank. A submodule of a free module need not be free, but over a principal ideal domain it is free.

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Further reading

  1. Web https://en.wikipedia.org/wiki/Free_module, Definition
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DefinesFree module
UsesModule over a ring

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