# Rendiconti del Seminario Matematico della Università di Padova

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**Volume 135, 2016, pp. 133–149**

**DOI: 10.4171/RSMUP/135-7**

Strongly FP-injective and strongly flat functors

Lixin Mao^{[1]}(1) Department of Mathematics and Physics, Nanjing Institute of Technology, 211167, Nanjing, Jiangsu, China

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.

*Keywords: *Strongly FP-injective functor, FP-injective functor, strongly flat functor, flat functor, (pre)cover, (pre)envelope

Mao Lixin: Strongly FP-injective and strongly flat functors. *Rend. Sem. Mat. Univ. Padova* 135 (2016), 133-149. doi: 10.4171/RSMUP/135-7