13 - The Artin-Rees Property

Summary

The Artin-Rees property is a condition with a long history in the theory of commutative noetherian rings (where every ideal satisfies the condition). Versions of this property have also played important roles in many verifications of the second layer condition, and they place certain restrictions on the possible structure of cliques of prime ideals. We introduce a convenient form of this property and some of its uses in this chapter, which is a continuation of Chapter 12. The reader may also treat this chapter as an appendix if desired, since the Artin-Rees property will not appear later in the text aside from a few exercises in the following chapter.

• THE ARTIN-REES PROPERTY •

Definition. An ideal I in a ring I has the right AR-property if, for every right ideal K of R, there is a positive integer n such that K ∩ Iⁿ ≤ KI. The left AR-property is defined symmetrically, and I has the AR-property if it has both the right and left AR-properties.

The reader should be warned that the definition just given is the weakest of several Artin-Rees properties discussed in the literature; in particular, in most of the commutative literature one finds a definition involving a stronger condition (see the proof of Lemma 13.2).