A module M is said to be weakly N-projective if it has a projective cover π: P (M)↠ M and for
each homomorphism: P (M)→ N there exists an epimorphism σ: P (M)↠ M such that (kerσ) …

ABSTRACT A module M Is said to be weakly projective If and only If It has a projective cover
11": P (M)--+ M and every map from P (M) Into a finitely generated (free) module can be …

The purpose of this paper is to initiate the study of improjective modules. A module X is im-
projective if given any epimorphism/: B-> A and any homomorphism/: X-> A, there exists a …

Given a module M over a ring R, G. Azumaya [1] introduced the dual notions of M-projective
and M-injective modules. These concepts have actually led MS Shrikhande to a study of …

Let R be a ring and M a fixed R-module. An R-module Q is called M-injective if, given a
monomorphism C $ of an R-module N into M, every homomorphism: N-+ Q can be extended …

The concept of weak relative-injectivity of modules was introduced originally in [10], where it
is shown that a semiperfect ring R is such that every cyclic right module is embeddable …

The structure of almost projective modules can be better understood in the case when the
following Condition (P) holds: The union of each countable pure chain of projective modules …

A ring $ R $ is regular [completely reducible] if and only if the character module of every left
$ R $-module is quasi-injective [quasiprojective]. Submodules of quasiprojective left $ R …

A module is called locally pure-projective if every pure epimor-phism onto it is locally split.
We prove that every locally pure-projective module is a Mittag-Leffler module. We derive …

The notation in this paper will be standard and it may be found in [2] or [8]. Throughout the
paper, the notation A⊂'B will mean that A is an essential submodule of the module B. Given …