  This paper is an exposition of the so-called injective Morita contexts (in
which the connecting bimodule morphisms are injective) and Morita
$\alpha$contexts (in which the connecting bimodules enjoy some local
projectivity in the sense of Zimmermann-Huisgen). Motivated by situations in
which only one trace ideal is in action, or the compatibility between the
bimodule morphisms is not needed, we introduce the notions of Morita
semi-contexts and Morita data, and investigate them. Injective Morita data will
be used (with the help of static and adstatic modules) to establish
equivalences between some intersecting subcategories related to subcategories
of modules that are localized or colocalized by trace ideals of a Morita datum.
We end up with applications of Morita $\alpha$-contexts to $\ast$-modules and
injective right wide Morita contexts.

Rings and Algebras