The large condition for rings with Krull dimension

Abstract

A module M with Krull dimension is said to satisfy the large condition if for any essential submodule L of M, the Krull dimension of M / L M/L is strictly less than the Krull dimension of M. For a right noetherian ring R with Krull dimension α \alpha this is equivalent to the condition that every f.g. uniform submodule of E ( R R ) E({R_R}) with Krull dimension α \alpha is critical. It is also shown that if R is right noetherian with Krull dimension α \alpha and if I 0 {I_0} is a right ideal maximal with respect to K dim ⁡ I 0 > α \dim {I_0} > \alpha , then R satisfies the large condition if and only if I 0 {I_0} is a finite intersection of cocritical right ideals and I 0 {I_0} is closed.