Ring satisfying PIT

This article defines a property of commutative unital rings; a property that can be evaluated for a commutative unital ring
View all properties of commutative unital rings
VIEW RELATED: Commutative unital ring property implications | Commutative unital ring property non-implications |Commutative unital ring metaproperty satisfactions | Commutative unital ring metaproperty dissatisfactions | Commutative unital ring property satisfactions | Commutative unital ring property dissatisfactions

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 domains, called Krull domains, 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 $R$ is said to satisfy PIT or satisfy the Principal Ideal Theorem if given $x\in R$, and given $P$ as a prime ideal minimal amoung the primes containing $x$, the codimension of $P$ is at most 1.

Relation with other properties

Stronger properties

• Noetherian ring: For full proof, refer: 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