Given $\displaystyle (\mathbb{Z}_n, +)$ and a new element $\displaystyle k$, for what $\displaystyle n$ can $\displaystyle k\cup(\mathbb{Z}_n, +)$ be turned into a ring so that + in $\displaystyle \mathbb{Z}_n$ becomes multiplication in $\displaystyle \mathbb{R}$, $\displaystyle k$ becomes $\displaystyle 0_\mathbb{R}$?