Definition:Generated Submodule

Definition

Let $R$ be a ring.

Let $M$ be an $R$-module.

Let $S\subset M$ be a subset of $M$.

$R$-module

The submodule generated by $S$ is the intersection of all submodules of $M$ containing $S$.

Unitary $R$-Module

Let $R$ be a ring with unity.

Let $M$ be a unitary $R$-module.

The submodule generated by $S$ is the set of all linear combinations of elements of $S$.

Linear Span in Vector Space

Let $K$ be a division ring or a field.

Let $V$ be a vector space over $K$.

Let $A \subseteq V$ be a subset of $V$.

Then the linear span of $A$, denoted $\span A$ or $\map \span A$, is the set of all linear combinations (of finite length) of vectors in $A$.

Also see

• Equivalence of Definitions of Generated Submodule over Ring with Unity
• Definition:Generator of Module
• Generated Submodule may not equal Set of Linear Combinations