Let R be a commutative ring with identity. A short exact sequence E of R-modules is said to
be neat (respectively, P-pure) if the sequence Hom R (S, E)(respectively, S⊗ RE) is exact for …

Let R be a commutative ring. We determine the cyclic modules in the projectivity domain and
those in the injectivity domain of a simple R-module. An R-module M is called poor (p-poor) …

PM Cohn [7] calls a submodule P of the left A-module M pure iff 0→ E⊗ P→ E⊗ M is exact for
all right modules E. This definition of purity, which Cohn [7] has shown to be equivalent to …

For a commutative local ring R, we know that every ideal is a direct sum of local modules if
and only if the unique maximal ideal M of R is of the form M= R x⊕ R y⊕(⊕ γ∈ Γ R s γ) …

R denotes a commutative ring. After Bass [B], a ring R is perfect in case every module has a
projective cover. A ring R is a max ring provided that every nonzero i2-module has a …

Let R be a ring. A right R-module M is said to be neat-flat if the kernel of any epimorphism Y-
> M is neat in Y, ie, the induced map Hom (S, Y)-> Hom (S, M) is surjective for any simple …

Let R be a ring. A right R-module M is said to be neat-flat if the kernel of any epimorphism
Y→ M is neat in Y, ie, the induced map Hom (S, Y)→ Hom (S, M) is surjective for any simple …

Let R be a ring with an identity element. We prove that R is right Kasch if and only if injective
hull of every simple right R-modules is neat-flat if and only if every absolutely pure right R …

We show that a commutative ring\(R\) has neat range one if and only if every unit modulo
principal ideal of a ring lifts to a neat element. We also show that a commutative morphic …

An R-module F is called closed (neat) projective if, for every closed (neat) submodule L of
every R-module M, every homomorphism from F to M/L lifts to M. In this paper, we study …