The question is as indicated in the title:
When is $\langle m+n{\Bbb Z}\rangle$ a radical ideal in ${\Bbb Z}_n$, i.e. $Rad(\langle m+n{\Bbb Z}\rangle)=\langle m+n{\Bbb Z}\rangle$?
I gathered the information in the following question I asked:
- When do we have $Rad(I)=I$ for an ideal $I$ of a ring $R$?
- How can I calculate the radical of an ideal in ring ${\Bbb Z}_n$?
According answers to the second question, I am able to get $ Rad(\langle m+n{\Bbb Z}\rangle)=\langle \bar{m}+n{\Bbb Z}\rangle$ for some $\bar{m}$. To determine whether $Rad(\langle m+n{\Bbb Z}\rangle)\subset\langle m+n{\Bbb Z}\rangle$, it suffices to know when $ \langle \bar{m}+n{\Bbb Z}\rangle\subset \langle m+n{\Bbb Z}\rangle $ Is this equivalent to $\langle \bar{m}\rangle\subset \langle m\rangle$ in $\Bbb Z$? How can I approach the problem in the title?