Example (2) on page 682 of Dummit and Foote reads as follows: (see attached)

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(2) For any field k, the ideal (x) in k[x,y] is primary since it is a prime ideal.

For any , the ideal is primary

since it is a power of the maximal ideal (x,y)

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My first problem with this example is as follows:

How can we demonstrate the the ideal (x,y) in k[x,y] is maximal

Then my second problem with the example is as follows:

How do we rigorously demonstrate that the ideal is primary.

D&F say that this is because it is the power of a maximal ideal - but where have they developed that theorem/result?

The closest result they have to that is the following part of Proposition 19 (top of page 682 - see attachment)

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Proposition 19. Let R be a commutative ring with 1

... ...

(5) Suppose M is a maximal ideal and Q is an ideal with

for some .

Then Q is a primary ideal with rad Q = M

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Now if my suspicions are correct and Proposition 19 is being used, then can someone explain (preferably demonstrate formally and rigorously)

how part (5) of 19 demonstrates that the ideal is primary on the basis of being a power of a maximal ideal.

Would appreciate some help

Peter