Zero Vector Space Product iff Factor is Zero

Theorem
Let $\left({G, \circ}\right)$ be a group whose identity is $e$.

Let $\left({K, +, \circ}\right)$ be a division ring whose zero is $0$ and whose unity is $1$.

Let $\left({G, +_G, \circ}\right)_K$ be a $K$-vector space.

Let $x \in G, \lambda \in K$.

Then $\lambda \circ x = e \iff \left({\lambda = 0 \lor x = e}\right)$.

Also see

 * Basic Results about Modules
 * Basic Results about Unitary Modules