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

Proof
A vector space is a module, so all results about modules also apply to vector spaces.

Suppose $$\lambda \circ x = e$$ but $$\lambda \ne 0$$.

Then from Basic Results about Modules $$(1)$$:

$$ $$ $$ $$ $$