In the present paper, we initiate a study
of a concept of amenability for discrete semigroups $S$ with
identity called strict inner amenability. We first study the
relations between the concepts inner amenability, strict inner
amenability, and amenability of $S$. We then introduce some
suitable tools and properties to give several characterizations of
strict inner amenability of $S$ such as Reiter's condition and
F$\o$lner's condition. We also offer a characterization of strict
inner amenability in terms of a fixed point property. As
applications of these results to discrete groups $G$, we obtain a
number of equivalent statements describing strict inner
amenability of $G$.