## Six-element vector space

Is it possible for the structure $$(\{0{,}1,2{,}3,4{,}5\},\oplus,\odot)$$ to form a vector space over the field $$\mathbb Z_3$$, where $$\mathbf u \oplus \mathbf v = \mathbf u+\mathbf v \mod 6$$ and $$a\odot \mathbf u= a\cdot \mathbf u \mod 6$$?
It is impossible, because for example $$2\odot 3=0$$ which is in conflict with axioms of a vector space.