The global dimension of FBN rings with enough clans

Abstract

For an FBN ring R R , a classical set of prime ideals { P 1 , … , P n } \left \{ {{P_1}, \ldots ,{P_n}} \right \} is one for which the semiprime ideal N = ∩ i = 1 n P i N = \cap _{i = 1}^n{P_i} satisfies the Artin-Rees property. A minimal classical set is called a clan. We say an FBN ring R R has enough clans if each prime ideal P P is an element of a clan. In this paper, we show that for such rings the Krull dimension is less than or equal to the global dimension.