Jacobson radical of $\mathbb Z/n\mathbb Z$, $J(\mathbb Z/n\mathbb Z)$

Question

It is clear that if $p_1, p_2, \dots, p_k$ are all the distinct prime divisors of $n$ then maximal ideals of $\mathbb{Z}/n\mathbb{Z}$ are $p_i \mathbb{Z}/n\mathbb{Z}$ as the quotient field will be $\mathbb{Z}/p_i\mathbb{Z}$ which is field. Now, I am interested to find the intersection of all of these maximal ideals.

$$J(R) = \bigcap p_i \mathbb{Z}/n\mathbb{Z}$$

However I couldn't manage to find. I saw one more claim which is $\bigcap p_i \mathbb{Z}/n\mathbb{Z}=p_1p_2\dots p_n \mathbb Z /n\mathbb Z$. How can I show this argument.

Answer 1

Let $R$ be a commutative ring, and let $I,J\subseteq R$ be ideals.

If $I+J=R$ then $$I\cap J=IJ$$

Proof: It is clear that $IJ\subseteq I\cap J$, to show the reverse inclusion notice that $$\begin{align}I\cap J&\subseteq (I\cap J)I+(I\cap J)J\\&\subseteq IJ\end{align}$$