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DVRs II August 30, 2009

Posted by Akhil Mathew in algebra, algebraic number theory, commutative algebra, number theory.
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Earlier I went over the definition and first properties of a discrete valuation ring.  Today, it’s time to say how we can tell a ring is a DVR–it turns out to be not too bad, which is nice because the properties we need in this criterion are often easier to work with than the existence of some discrete valuation.

Today’s result is:

Theorem 1 If the domain {R} is Noetherian, integrally closed, and has a unique nonzero prime ideal {\mathfrak{m}}, then {R} is a DVR. Conversely, any DVR has those properties. (more…)

A prime ideal criterion for being Noetherian August 13, 2009

Posted by Akhil Mathew in algebra, commutative algebra.
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This post, the third in the mini-series so far, gives one more criterion for when a ring is Noetherian.  I also discuss how prime ideals tend to crop up in commutative algebra.

 Why prime ideals are important

 As discussed in the end of my previous post and in the comments, ideals satisfying some property and maximal with respect to it are often prime. To prove these results, we often use the following convenient notation: 

Definition 1

If {I,J} are ideals of a commutative ring {A}, then we define