I assume that conditions 1) and 2) are applies to nonzero $z$ only, otherwise each of them ia absurd.
Choose a base in $\mathbb R^n$ so that all nonzero elements of $\mathbb Z^n$ have nonzero yet algebraic coordinates. (It suffices to choose a base consisting of vectors whose coordinates are $\mathbb Q$-independent algebraic numbers). In a new base, refine a coordinatewise multiplication $*$. Finally, define $x\circ y=(ex)*y$. The operation $\circ$ is what you need since no nonzero vector in $\mathbb Z^n$ is a product of two other such vectors.