Algebras for which every indecomposable right module is invariant in its injective envelope

The algebras of the title are characterized as those having Tachikawa's strong left cyclic representation type (SLCRT) with the further property that every quotient of a principal indecomposable left module has square free socle. Moreover it turns out that these are precisely the algebras for which every indecomposable right module is quasi-injective in the sense of Johnson and Wong, and that these algebras have finite module type.