On f-injective modules

In this paper, the notions of f-injective and f^*-injective modules are introduced. Elementary properties of these modules are given. For instance, a ring R is coherent iff any ultraproduct of f-injective modules is absolutely pure. We prove that the class S^* Sigma^* of f^*-injective modules is closed under ultraproducts. On the other hand, S^* Sigma^* is not axiomatisable. For coherent rings R, S^* Sigma^* is axiomatisable iff every c₀ chi_0 -injective module is f^*-injective. Further, it is shown that the class S Sigma of f-injective modules is axiomatisable iff R is coherent and every c₀ chi_0 -injective module is f-injective. Finally, an f-injective module H, such that every module embeds in an ultraprower of H, is given.