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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:

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?

1 Answers 1