In a commutative ring, prime ideals are radical

Let R be a commutative ring. An ideal I of R is called radical if \mathsf{rad}(I) = I (since I \subseteq \mathsf{rad}(I) always holds, we can say equivalently that I is radical if \mathsf{rad}(I) \subseteq I).

  1. Prove that every prime ideal of R is radical.
  2. Let n > 1 be an integer. Prove that 0 is radical in \mathbb{Z}/(n) is radical if and only if n is squarefree. Deduce that (n) is radical in \mathbb{Z} if and only if n is squarefree.

We begin with a lemma.

Lemma: Let R be a commutative ring and let I \subseteq R be an ideal. Also let J \subseteq R is an ideal containing I. Then J/I is radical in R/I if and only if J is radical in R. Proof: Suppose J/I is radical in R/I, and let x \in \mathsf{rad}(J). Then x^m \in J for some m \geq 1. Now (x+I)^m = x^m+I \in J/I, so that x+I \in \mathsf{rad}(J/I) = J/I. Thus x \in J, and J is a radical ideal in R. Conversely, suppose J is radical in R and let x+I \in \mathsf{rad}(J/I). Then (x+I)^m = x^m+I \in J/I for some m \geq 1, so that x^m \in J. Thus x \in \mathsf{rad}(J) = J, and we have x+I \in J/I; hence J/I is radical. \square

  1. Let P \subseteq R be a prime ideal. Let x \in \mathsf{rad}(P). Then for some n \geq 1, we have x^n \in P; suppose that n is minimal with this property. If n \geq 2, then xx^{n-1} \in P. Since P is prime, then either x \in P or x^{n-1} \in P, a contradiction since n is minimal. Thus n = 1, and we have x \in P.
  2. Write the prime factorization of n as n = \prod p_i^{k_i}. Suppose first that 0 is radical in \mathbb{Z}/(n), and suppose some k_i is at least 2. Now \prod p_i is nonzero in \mathbb{Z}/(n), and (\prod p_i)^{\max k_i} = 0, a contradiction. Thus n is squarefree. Conversely, suppose n is squarefree and let a \in \mathbb{Z}/(n) with a^k = 0 for some k. If some prime p divides n but not a, then no power of a can be divisible by n, a contradiction. Thus n divides a, and in fact \mathsf{rad}(0) = 0.

    Consider (n) as an ideal of \mathbb{Z}. By the lemma, (n) is radical in \mathbb{Z} if and only if 0 is radical in \mathbb{Z}/(n), if and only if n is squarefree.

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