Title:  (Flat) modules that are fully invariant in their pure-injective (cotorsion) envelopes

Abstract:  We introduce two concepts. A module M is said to be purely quasi-injective (resp. quasi-cotorsion) if it is fully invariant in its pure-injective envelope (resp. if it is flat and fully invariant in its cotorsion envelope). Endomorphism rings of both of the above types of modules are proved to be regular and self injective modulo their Jacobson radicals. If M is a purely quasi-injective (resp. quasi-cotorsion) module, then so is any finite direct sum of copies of M. Each of the above concepts is stronger than the well-known concept of quasi-pure-injectivity, but not equivalent. This solves, negatively, a problem of Mao and Ding's of whether every flat quasi pure-injective module is fully invariant in its cotorsion envelope. Certain types of rings are characterized in terms of purely quasi-injective modules. For example, a ring R is regular if and only if every purely quasi-injective R-module is quasi-injective, and is pure-semisimple if and only if every R-module is purely quasi-injective.