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