An approach to Boyle's conjecture

Abstract

A ring R is called a right QI-ring if every quasi-injective right R-module is injective. The well-known Boyle's Conjecture states that any right QI-ring is right hereditary. In this paper we show that if every continuous right module over a ring R is injective, then R is semisimple artinian. In fact, if every singular continuous right R-module satisfying the restricted semisimple condition is injective, then R is right hereditary. Moreover, in this case, every singular right R-module is injective.