Artinian quotient rings of ideal invariant noetherian rings

Abstract

A ring R is ideal invariant on the right if for any two ideals 5’ and T the right Krull dimension ( SITS JR of the bimodule SITS is less than or equal to the right Krull dimension of R/T. This condition was introduced by Warfieldl, it is a weaker form of the concept of right ideal invariance due to Stafford [15] which requires the above inequality to hold whenever T is a right ideal. In recent years, several conditions have emerged which assure the existence of artinian classical quotient rings for various types of (left and right) noetherian rings. One of these is K-homogeneity, a condition which requires the Krull dimension 1 A jR of each non-zero right ideal A of the ring R to be equal to j R lR . The most general theorem in this connection appears in [8], it establishes the existence of a right artinian quotient ring Q(R) for any K-homogeneous ring R with right Krull dimension whose nil radical N is weakly right ideal invariant, meaning that N satisfies ) N/AN IR < ) R/N IR for any right ideal AwithjR/Aj<jR/NI,. Two previously known theorems are special cases of this: Gordon’s result [5] for FBN rings and Lenagan’s [lo] for noetherian rings with Krull dimension one. The assumption of R being K-homogeneous is by no means necessary for the existence of an artinian quotient ring. In general, the largest ideal with Krull dimension smaller than that of R does not even have to be a direct summand.