nLab
basis of a free module

Bases of a free module

Definition

A basis of a free RR-module MM (possibly a vector space, see basis of a vector space) is a linear isomorphism B:M iIRB\colon M \to \oplus_{i\in I}R to a direct sum of copies of the ring RR, regarded as a module over itself.

We see how this is equivalent to the classical definition of a basis as a linearly independent spanning set:

Lemma

A basis for a free RR-module MM determines a unique generating set for MM of linearly independent elements of MM.

Proof

Fix a basis BB for MM over RR. Then let a i(δ ij) jIa_i\coloneqq (\delta_{ij})_{j\in I} for each iIi\in I. Since BB is an isomorphism, each a ia_i determines a unique element b iB 1(a i)b_i \coloneqq B^{-1}(a_i). Since every element of MM is of the form B 1(x)B^{-1}(x) for x iIRx\in \oplus_{i\in I}R, and since every element of iIR\oplus_{i\in I}R can be written as a finite RR-linear combination of the a ia_i, this proves that {b i} iI\{b_i\}_{i\in I} generates MM. To show linear independence, we again apply BB and its linearity. The result is immediate.

Revised on October 23, 2012 00:54:20 by Urs Schreiber (82.169.65.155)