On infinite direct sums of lifting modules

Abstract

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