  Let $R$ be a ring and $S$ a multiplicative subset of $R$. An $R$-module $P$
is called uniformly $S$-projective provided that the induced sequence
$0\rightarrow \mathrm{Hom}_R(P,A)\rightarrow \mathrm{Hom}_R(P,B)\rightarrow
\mathrm{Hom}_R(P,C)\rightarrow 0$ is $u$-$S$-exact for any $u$-$S$-short exact
sequence $0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$. Some
characterizations and properties of $u$-$S$-projective modules are obtained.
The notion of $u$-$S$-semisimple modules is also introduced. A ring $R$ is
called a $u$-$S$-semisimple ring provided that any free $R$-module is
$u$-$S$-semisimple. Several characterizations of $u$-$S$-semisimple rings are
provided in terms of $u$-$S$-semisimple modules, $u$-$S$-projective modules,
$u$-$S$-injective modules and $u$-$S$-split $u$-$S$-exact sequences.
