LAUSR

Let $R$ be a ring and let $S$ be a multiplicative subset of $R$ . An $R$ -module $M$ is said to be a $u$ - $S$ -absolutely pure module if ${\text{Ext}}_R^1\left(N,M\right)$ is $u$ - $S$ -torsion for any finitely presented $R$ -module $N$ . This paper introduces and studies the notion of $S$ -FP-projective modules, which extends the classical notion of FP-projective modules. An $R$ -module $M$ is called an $S$ -FP-projective module if ${\text{Ext}}_R^1\left(M,N\right)=0$ for any $u$ - $S$ -absolutely pure $R$ -module $N$ . We also introduce the $S$ -FP-projective dimension of a module and the global $S$ -FP-projective dimension of a ring. Then, the relationship between the $S$ -FP-projective dimension and other homological dimensions is discussed.