In \cite[Theorem 3.1]{oiio} K. R. Fuller characterized indecomposable injective projective modules over artinian rings using $i$-pairs. In \cite{ioqm} the author generalized this theorem to indecomposable projective quasi-injective modules and indecomposable quasi-projective injective modules over artiniain rings. In \cite{oato} the author and K. Oshiro studied the above Fuller's theorem minutely. Further in \cite{ipip}, \cite{cpip} M. Hoshino and T. Sumioka extended these results to perfect rings. In this paper we studies the results in \cite{ioqm} from the point of view of \cite{oato}, \cite{ipip}.