Definition:Basis of Module

Definition

Let $R$ be a ring with unity.

Let $\struct {G, +_G, \circ}_R$ be a unitary $R$-module.

Definition 1

A basis of $G$ is a linearly independent subset of $G$ which is a generator for $G$.

Definition 2

Let $\BB = \family {b_i}_{i \mathop \in I}$ be a family of elements of $M$.

Let $\Psi: R^{\paren I} \to M$ be the homomorphism given by Universal Property of Free Module on Set.

Then $\BB$ is a basis if and only if $\Psi$ is an isomorphism.

Also see

• Equivalence of Definitions of Basis of Module
• Definition:Free Module
• Definition:Dimension of Module
• Bases of Free Module have Equal Cardinality
• Definition:Ordered Basis

• Results about bases of modules can be found here.

Special cases

• Definition:Basis of Vector Space

Linguistic Note

The plural of basis is bases.

This is properly pronounced bay-seez, not bay-siz.