This paper is supplemental to [4],[6] and [7]. We shall show, under assumption of finite
length, that when we study almost relative projectives, we may restrict ourselves only to …

Recently,]~. DE ROBERT has considered the dual concepts of relative projectivity and
relative injectivity that extend the notions of quasi-projective and quasi-injective modules …

We have defined a new concept of almost M-projectives [7] and given several properties of
them [4]. This paper is a continuous work of [4] and [7]. If a module MQ is Mrprojective for a …

We have defined a new concept of almost relative projectivity [7], If a module Mo is M,-
ρrojective for a finite set of modules Miy then Mo is 2,-φ Λff-projective [2]. However this fact is …

Introduction and notation. Throughout this paper all rings are associative rings with identity
and all modules are unitary left modules. An R-module M is quasi-injective in case HomR …

If M denotes the class of unitary left modules over a ring R, then each subclass# ofM
determines two subclasses ofJt—the relative projectivity class and the relative injectivity …

The restricted injective envelopes with respect to various types of preradicals have been
studied by many authors (see eg [2],[6],[8],[10]). In the second paragraph we shall dualize …

M commutes. The first section of this paper contains results which are needed to prove a
decomposition theorem for quasi-projective modules over left perfect rings (Theorem 1.10) …

We have introduced a concept of almost relative projectives (resp. injectives) in [7](resp.[2])
which is deeply related with lifting modules [9](resp. extending modules [10]). When we …

We have defined aconcept of almost M-projectives and almost M-injectives in [4] and [9],
respectively. In the first section of this paper we give some relations among lifting modules …