Maximal Radical implies Primary Ideal
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Theorem
Let $R$ be a commutative ring with unity.
Let $\mathfrak a$ be an ideal of $R$.
Let $\map \Rad {\mathfrak a}$ be the radical of $\mathfrak a$.
Suppose that $\map \Rad {\mathfrak a}$ is a maximal ideal.
Then $\mathfrak a$ is a primary ideal.
Proof
Consider the quotient ring $R / \mathfrak a$.
By Definition 1 of Nilradical of Ring and Definition 1 of Radical of Ideal of Ring:
- $\Nil {R / \mathfrak a} = \map \Rad {\mathfrak a} / \mathfrak a$
On the other hand, by hypothesis, $\map \Rad {\mathfrak a} / \mathfrak a$ is a maximal ideal of $R / \mathfrak a$.
Thus, in view of Definition 1 of Nilradical of Ring:
- $\Nil {R / \mathfrak a}$
is the only maximal ring in $R / \mathfrak a$.
In view of Proper Ideal of Ring is Contained in Maximal Ideal, Definition 2 of Primary Ideal follows.
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