Ring satisfying PIT - Revision history

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2008-05-12T16:34:34Z

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2008-01-07T23:50:05Z

References

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	==References==		==References==

	* ''On the generalized principal ideal theorem and Krull domains'' by David F. Anderson, David E. Dobbs, Paul M. Eakin, Jr. and William J. Heinzer		* ''On the generalized principal ideal theorem and Krull domains'' by David F. Anderson, David E. Dobbs, Paul M. Eakin, Jr. and William J. Heinzer, ''Pacific Journal of Mathematics, Vol. 146, No. 2, 1990''

Vipul at 23:49, 7 January 2008

2008-01-07T23:49:11Z

New page

{{curing property}}

==History==

===Pre-origin===

[[Krull's principal ideal theorem]], often abbreviated as the PIT, is a statement about a property (condition) that Noetherian rings satisfy. Krull also came up with a class of [[integral domain]]s, called [[Krull domain]]s, which satisfy the property.

===Origin===

A study of ''rings satisfying PIT'' was undertaken, for instance, in the paper ''On the generalized principal ideal theorem and Krull domains'' by Anderson, Dobbs, Eakin and Heinzer.

==Definition==

===Definition with symbols===

A [[commutative unital ring]] <math>R</math> is said to '''satisfy PIT''' or '''satisfy the Principal Ideal Theorem''' if given <math>x \in R</math>, and given <math>P</math> as a prime ideal minimal amoung the primes containing <math>x</math>, the codimension of <math>P</math> is at most 1.

==Relation with other properties==

===Stronger properties===

* [[Noetherian ring]]: {{proofat|[[Krull's principal ideal theorem]]}}
* [[Krull domain]]
* [[Ring satisfying GPIT]]

==References==

* ''On the generalized principal ideal theorem and Krull domains'' by David F. Anderson, David E. Dobbs, Paul M. Eakin, Jr. and William J. Heinzer