Definition:Trivial Module

Definition

Let $\struct {G, +_G}$ be an abelian group whose identity is $e_G$.

Let $\struct {R, +_R, \circ_R}$ be a ring.

Let $\circ$ be defined as:

$\forall \lambda \in R: \forall x \in G: \lambda \circ x = e_G$

Then $\struct {G, +_G, \circ}_R$ is an $R$-module.

Such a module is called a trivial module.

Also see

• Trivial Module is Module
• Trivial Module is Not Unitary

• Definition:Zero Module

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 26$. Vector Spaces and Modules: Example $26.6$