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- #1

- Jun 22, 2012

- 2,918

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The primary ideals in [TEX] \mathbb{Z} [/TEX] are 0 and the ideals [TEX] (p^m) [/TEX] for p a prime and [TEX] m \ge 1 [/TEX].

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So given what D&F say, (4) is obviously not primary.

I began trying to show from definition that (4) was not a primary from the definition, but failed to do this

**Can anyone help in this ... and come up with an easy way to show that (4) is not primary?**

Further, can anyone please help me prove that the primary ideals in [TEX] \mathbb{Z} [/TEX] are 0 and the ideals [TEX] (p^m) [/TEX] for p a prime and [TEX] m \ge 1 [/TEX].

Further, can anyone please help me prove that the primary ideals in [TEX] \mathbb{Z} [/TEX] are 0 and the ideals [TEX] (p^m) [/TEX] for p a prime and [TEX] m \ge 1 [/TEX].

Peter

Note: the definition of a primary idea is given in D&F as follows:

Definition. A proper ideal Q in the commutative ring R is called primary if whenever [TEX] ab \in Q [/TEX] and [TEX] a \notin Q [/TEX] then [TEX] b^n \in Q [/TEX] for some positive integer n.

Equivalently, if [TEX] ab \in Q [/TEX] and [TEX] a \notin Q [/TEX] then [TEX] b \in rad \ Q [/TEX]