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 $\displaystyle n \ge 1 $, the ideal $\displaystyle (x,y)^n $ 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 $\displaystyle (x,y)^n $ 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 $\displaystyle M^n \subseteq Q \subseteq M $

for some $\displaystyle n \ge 1$.

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 $\displaystyle (x,y)^n $ is primary on the basis of being a power of a maximal ideal.

Would appreciate some help

Peter