Category:Definitions/Generators of Modules

This category contains definitions related to Generators of Modules.
Related results can be found in Category:Generators of Modules.

Let $R$ be a ring.

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

Let $S \subseteq M$ be a subset.

Definition 1

$S$ is a generator of $M$ if and only if $M$ is the submodule generated by $S$.

Definition 2

$S$ is a generator of $M$ if and only if $M$ has no proper submodule containing $S$.

Generator of Unitary Module

Let $R$ be a ring with unity.

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

Let $S \subseteq M$ be a subset.

$S$ is a generator of $M$ if and only if every element of $M$ is a linear combination of elements of $S$.