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 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.