Definition:Basis of Module

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Let $R$ be a ring with unity.

Let $\left({G, +_G, \circ}\right)_R$ 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 $\mathcal B = \family {b_i}_{i \mathop \in I}$ 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 $\mathcal B$ is a basis if and only if $\Psi$ is an isomorphism.

Also see

  • Results about bases of modules can be found here.

Special cases

Linguistic Note

The plural of basis is bases.

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