On infinite direct sums of lifting modules

The aim of the present article is to investigate the structure of rings R satisfying the condition: for any family {Si|i ∈ ℕ} of simple right R-modules, every essential extension of ⊕i∈ℕE(Si) is a direct sum of lifting modules, where E(−) denotes the injective hull. We show that every essential extension of ⊕i∈ℕE(Si) 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 Σ-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 Σ-V-ring.