Rings where the annihilators of<i>α</i>-critical modules are prime ideals

Abstract

For a ring R with Krull dimension a, we investigate the property that the annihilators of -critical modules are prime ideals. If R satisfies the large condition then this property holds iff RII 0 is semiprime, where J o is the maximal right ideal of Krull dimension <a. The property holds in the following rings, (i) R is weakly ideal invariant, (ii) R satisfies the left AR property, or (iii) the prime ideals of R are right localizable. In addition, if R is a hereditary Noetherian ^-primitive ring, then R is a prime ring.