Strictness of STT-injectivity versus pure-injectivity

Determine whether there exists a ring R for which the class of STT-injective R-modules strictly contains the class of pure-injective R-modules, i.e., whether injectivity with respect to all inclusions of the form N ∩ A ↪ A with A an R-module and N ≺ (H_θ, ∈) containing R (the class 𝒮𝒯𝒯_R) is strictly weaker than pure-injectivity; equivalently, decide whether the containment {pure-injective R-modules} ⊆ {𝒮𝒯𝒯_R-injective modules} is strict for some ring R.

Background

In proving the Eklof–Trlifaj theorem, the paper observes that one can weaken the usual hypothesis that every B in a class ℬ is pure-injective to the weaker requirement that each B is injective only with respect to a special family of pure embeddings: inclusions of the form N ∩ A ↪ A where A ∈ N ≺ (H_θ, ∈) and R ⊆ N. This motivates defining 𝒮𝒯𝒯_R as the class of such inclusions.

While every such inclusion is pure, not every pure embedding arises in this way. It is therefore natural to ask whether injectivity with respect to 𝒮𝒯𝒯_R coincides with classical pure-injectivity or is strictly weaker. Establishing a strict separation (or equality) would clarify whether the proof technique genuinely extends the Eklof–Trlifaj theorem beyond the regime of pure-injectives.