Strongly FP-injective and strongly flat functors

Abstract

A functor $F\in$ (mod-$R$, Ab) is called strongly FP-injective if $F$ is isomorphic to some functor $-\otimes M$ in (mod-$R$, Ab) with $M$ an FP-injective left $R$-module. A functor $G\in$ ((mod-$R{)}^{\mathrm{op}}$ Ab) is said to be strongly flat if $G$ is isomorphic to some functor $(-,N)$ in ((mod-$R{)}^{\mathrm{op}}$, Ab) with $N$ a flat right $R$-module. We study the properties of strongly FP-injective functors and explore the relationship between strongly FP-injective functors and strongly flat functors. Precovers and preenvelopes by strongly FP-injective and strongly flat functors are also investigated.