On infinite direct sums of lifting modules

Abstract

The aim of the present article is to investigate the structure of rings R satisfying the condition: for any family {S-i vertical bar i is an element of N} of simple right R-modules, every essential extension of circle plus E-i is an element of(S-i) is a direct sum of lifting modules, where E(-) denotes the injective hull. We show that every essential extension of circle plus E-i is an element of N(S-i) is a direct sum of lifting modules if and only if R is right Noetherian and E(S) is hollow. Assume that M is an injective right R-module with essential socle. We also prove that if every essential extension of M-(N) is a direct sum of lifting modules, then M is Sigma-injective. As a consequence of this observation, we show that R is a right V-ring and every essential extension of S-(N) is a direct sum of lifting modules for all simple modules S if and only if R is a right Sigma-V-ring.