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Let $R$ be the ring $\mathbb Z_{36}$.

How can I calculate $ \sqrt{\langle 0\rangle} , \sqrt{\langle 9\rangle} $?

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$\DeclareMathOperator{\Z}{\mathbf{Z}}$You're calculating radicals in $\Z/36\Z$. Ideals in that ring correspond to ideals of $\Z$ which contain $36$, and this correspondence preserves the property of an ideal being prime. Remember that the radical of an ideal is equal intersection of all prime ideals containing it. So, for example, which two prime ideals of $\Z$ contain $36$?

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If $\mathfrak m_1,\dots,\mathfrak m_n$ are distinct maximal ideals of a Dedekind domain, and if $k_1,\dots,k_n$ are positive integers, then the radical of $$ \prod_{i=1}^n\ \mathfrak m_i^{k_i} $$ is $$ \prod_{i=1}^n\ \mathfrak m_i. $$

EDIT. The following fact is also relevant. If $\mathfrak a$ is an ideal of a commutative ring $A$, then the correspondence between ideals of $A/\mathfrak a$ and ideals of $A$ containing $\mathfrak a$ is compatible with radicals.

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