Radical Ideal iff Quotient Ring is Reduced

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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 if and only if the quotient ring $R / J$ is a reduced ring.


Proof

Since $J \subset R$, it follows from:

Quotient Ring of Commutative Ring is Commutative

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 zero of $R / J$.


Sufficient Condition

Let $J$ be radical.

We need to show that:

if $x + J \in R / J$ such that $\left({x + J}\right)^n = 0_{R / J}$ for some positive integer $n$
then $x + J = 0_{R / J}$.


Let $\left({x + J}\right)^n = 0_{R / J}$.

Then:

$x^n + J = 0_{R / J}$

and therefore:

$x^n \in J$

Because $J$ is radical it follows that:

$x \in J$

That is:

$x + J = 0_{R / J}$

Therefore $R / J$ is reduced.

$\Box$


Necessary Condition

Let $R / J$ be reduced.

Let $x \in R$ be such that $x^n \in J$.

Then:

$0_{R / J} = J = x^n + J = \left({x + J}\right)^n$

Because $A / J$ is reduced, this implies that:

$x + J = 0_{R / J}$

That is:

$x \in J$

This shows that $J$ is radical.

$\blacksquare$