The concept of pure injectivity has played a central role in model theory of modules for the last five decades. Pure injective modules, also called algebraically compact, represent (besides other things) the sufficiently saturated models in the theory. They are close counterparts of pure embeddings: an embedding f is pure if and only if Hom(f,P) is surjective for each pure-injective module P.

Much less has been known about their natural infinitary generalisations - $\lambda$-pure-injective modules, for an uncountable regular cardinal $\lambda$. Although examples of nontrivial $\lambda$-pure embeddings are rather prevalent, their counterparts emerged extremely sparsely in the literature; mostly because of the lack of nontrivial, i.e., non-pure-injective examples.

We show that this was no coincidence: even though nontrivial examples of $\lambda$-pure-injectives naturally exist under some large cardinal assumptions (and they can even play a useful role here and there), without zero sharp, $\lambda$-pure-injectivity reduces to pure-injectivity for any uncountable regular $\lambda$.