# Sequence of Prime ideals in a Polynomial Ring

In the $d$ variable polynomial ring $R=k[x_{1},\cdots,x_{d}]$ show that $0, x_{1}R, (x_{1},x_{2})R, \cdots , (x_{1},x_{2},...,x_{d})R$ is a strictly increasing sequence of prime ideals and there is no longer such chain. how do i prove this claim.

Well, i am thinking that for this problem this result may be helpful: for a ring, $R$, to be Noetherian, one formulation dictates that any ascending chain of ideals in R terminates. But i just can't get going with it.

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A standard proof uses Noether normalization. For a superb introduction to (Krull) dimension of rings see Chapter 8, "Introduction to dimension theory" in Eisenbud's "Commutative algebra, with a view toward algebraic geometry", esp. Section 8.2.1, and for the proof see Section 13.1.

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Is this a theorem or a problem from that book. –  anonymous Oct 4 '10 at 16:46
@Chandru1: It is a theorem. Further, the introductory Chapter 8 on dimension theory is by far the best in any textbook, so I highly recommend that you consult that first for motivation. –  Bill Dubuque Oct 4 '10 at 17:00

Bill is right that the most standard proof uses Noether normalization. Another proof exploiting the fact that $R$ is a Hilbert-Jacobson ring can be found in Section 8.2 of

http://math.uga.edu/~pete/integral.pdf

The exposition here follows Kaplansky's Commutative Rings.

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Thanks Pete, there is an $R$ above your post,i wonder why that has come. –  anonymous Oct 4 '10 at 18:20