On \(f\)-injective modules (Q1614898)

Two generalizations of injectivity are studied in this paper. An \(R\)-module \(X\) is said to be \(f\)-injective (resp. \(f^*\)-injective) if, given any monomorphism \(F\to Y\), where \(F\) is a finitely generated (resp. finitely presented) module, any homomorphism \(F\to X\) can be extended to a homomorphism \(Y\to X\). These notions are claimed to be ``introduced'' here, but both have been known for years: \(f\)-injectivity was introduced by \textit{V. S. Ramamurthi} and \textit{K. M. Rangaswamy} as finite injectivity [J. Aust. Math. Soc. 16, 239-248 (1973; Zbl 0267.16014)] and \(f^*\)-injectivity as \(fp\)-injectivity by \textit{S. Jain} [Proc. Am. Math. Soc. 41, 437-442 (1973; Zbl 0246.16013)]. Apart from the definitions and some elementary results which coincide, however, this paper differs completely from the earlier papers. The class \(\Sigma^*\) of \(f^*\)-injective modules is shown to be closed under ultraproducts, but it is not axiomatizable. For coherent rings any ultraproduct of \(f\)-injective modules is shown to be absolutely pure. Necessary and sufficient conditions for both the class \(\Sigma^*\) and the class \(\Sigma\) of \(f\)-injective modules to be axiomatizable are found. The paper concludes with a proposition showing that there is an \(f\)-injective \(R\)-module \(H\) such that every module embeds in an ultrapower of \(H\).