Integral Domain is Reduced Ring
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Theorem
Let $\left({D, +, \circ}\right)$ be an integral domain.
Then $D$ is reduced.
Proof
Let $x \in D$ be a nilpotent element.
Then by Nilpotent Element is Zero Divisor, $x$ is a zero divisor in $D$.
By the definition of an integral domain, this means that $x = 0$.
Therefore the only nilpotent element of $D$ is $0$.
That is, $D$ is reduced.
$\blacksquare$