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:
and:
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.
$\blacksquare$