Almost free modules, perfect decomposition and Enochs's conjecture

Given a module X and a regular cardinal κ we study various notions of (κ,Add(X))-freeness and (κ,Add(X))-separability. Bearing on appropriate set-theoretic assumptions, we construct a non-trivial κ⁺-generated, (κ⁺,Add(X))-free and (κ⁺,Add(X))-separable module. Our construction allows κ to be singular thus extending [Theorem~4.7]{CortesGuilTorrecillas}. Bearing on similar set-theoretic assumptions, we characterize when every module X has a perfect decomposition. As a subproduct we show that Enoch's conjecture for classes Add(X) is consistent with ZFC -- a fact first proved by {S}aroch {Saroch}.