Prove Addition Exists....Sorta

May 2010
1
0
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 \(\displaystyle 0_\mathbb{R}\)?