The cotorsion pair generated by the Gorenstein projective modules and λ-pure-injective modules

We prove that, if GProj is the class of all Gorenstein projective modules over a ring R, then $${\frak{G}\frak{B}}=(\text{GProj},\text{GProj}^{\bot})$$ G B = ( GProj , GProj ⊥ ) is a cotorsion pair. Moreover, $${\frak{G}\frak{B}}$$ G B is complete when all projective modules are λ-pure-injective for some infinite regular cardinal λ (in particular, if R is right Σ-pure-injective); the latter condition is shown to be consistent with the axioms of ZFC modulo the existence of strongly compact cardinals. We also thoroughly study λ-pure-injective modules for an arbitrary infinite regular cardinal λ, proving along the way that: any cosyzygy module in an injective coresolution of a λ-pure-injective module is λ-pure-injective; the cotorsion pair cogenerated by a class of λ-pure-injective modules is cogenerated by a set and, under an additional technical assumption, generated by a set. Finally, assuming the set-theoretic hypothesis that 0^♯ does not exist, we prove that the category of right R-modules has enough λ-pure-injective objects if and only if the ring R is right pure-semisimple. This, in turn, follows from a rather surprising result that λ-pure-injectivity amounts to pure-injectivity in the absence of 0^♯.