Radical Ideal iff Quotient Ring is Reduced

Theorem
Let $\left({R, +, \circ}\right)$ be a commutative ring with unity.

Let $J$ be an ideal of $R$.

Then $J$ is a radical ideal iff the quotient ring $R / J$ is a reduced ring.

Proof
Since $J \subset R$, it follows from Commutative Quotient Ring and Quotient Ring of Ring with Unity is Ring with Unity that $R / J$ is a commutative ring with unity.

Let $0_{R/J}$ be the additive identity of $R/J$.

First suppose that $J$ is radical.

We need to show that if $x+J\in R/J$ such that $(x+J)^n=0_{R/J}$ for some positive integer $n$, then $x+J=0_{R/J}$.

If $(x+J)^n=0_{R/J}$, then $x^n+J=0_{R/J}$, and therefore $x^n\in J$.

Because $J$ is radical it follows that $x\in J$, i.e. $x+J=0_{R/J}$.

Therefore $R/J$ is reduced.

Conversely, suppose that $A/J$ is reduced, and let $x\in R$ be such that $x^n\in J$.

Then:
 * $0_{R/J}=J=x^n+J=(x+J)^n$

Because $A/J$ is reduced, this implies that $x+J=0_{R/J}$, i.e. $x\in J$.

This shows that $J$ is radical.