On infinite direct sums of lifting modules

Abstract

© 2017 World Scientific Publishing CompanyThe aim of the present article is to investigate the structure of rings (Formula presented.) satisfying the condition: for any family (Formula presented.) of simple right (Formula presented.)-modules, every essential extension of (Formula presented.) is a direct sum of lifting modules, where (Formula presented.) denotes the injective hull. We show that every essential extension of (Formula presented.) is a direct sum of lifting modules if and only if (Formula presented.) is right Noetherian and (Formula presented.) is hollow. Assume that (Formula presented.) is an injective right (Formula presented.)-module with essential socle. We also prove that if every essential extension of (Formula presented.) is a direct sum of lifting modules, then (Formula presented.) is (Formula presented.)-injective. As a consequence of this observation, we show that (Formula presented.) is a right V-ring and every essential extension of (Formula presented.) is a direct sum of lifting modules for all simple modules (Formula presented.) if and only if (Formula presented.) is a right (Formula presented.)-V-ring.