Null Module is Module
Jump to navigation
Jump to search
Theorem
Let $\struct {R, +_R, \circ_R}$ be a ring.
Let $G$ be the trivial group.
Let $\struct {G, +_G, \circ}_R$ be the null module.
Then $\struct {G, +_G, \circ}_R$ is a module.
Proof
Follows from the fact that $\struct {G, +_G, \circ}_R$ has to be, by definition, a trivial module:
$\circ$ can only be defined as:
- $\forall \lambda \in R: \forall x \in G: \lambda \circ x = e_G$
$\blacksquare$