Recently, many authors have embraced the study of certain properties modules such as projectivity, injectivity and flatness from an alternative point view. Rather than saying a module has property or not, each is assigned relative domain which, somehow, measures to which extent it this particular property. In work, we introduce new fresh perspective on modules. However, will first investigate m...

It is widely recognized that steganography with sideinformation in the form of a precover at the sender enjoys significantly higher empirical security than other embedding schemes. Despite the success of side-informed steganography, current designs are purely heuristic and little has been done to develop the embedding rule from first principles. Building upon the recently proposed MiPOD stegano...

It is proved undecidable in ZFC + GCH whether every Z-module has a {Z}-precover. Let F be a class of R-modules of the form C = {A : Ext(A,C) = 0 for all C ∈ C} for some class C. The first author and Jan Trlifaj proved [7] that a sufficient condition for every module M to have an F -precover is that there is a module B such that F = {B} (= {A : Ext(B,A) = 0}). In [8], generalizing a method used ...

In this paper, we introduce and investigate the notions of ξ-strongly copure projective objects in a triangulated category. This extends Asadollahi’s notion of ξ-Gorenstein projective objects. Then we study the ξ-strongly copure projective dimension and investigate the existence of ξ-strongly copure projective precover.

In an attempt to alleviate the negative impact of unavailable cover model, some steganographic schemes utilize the knowledge of the so-called “precover” when embedding secret data. The precover is typically a higherresolution (unquantized) representation of the cover, such as the raw sensor output before it is converted to an 8-bit per channel color image. The precover object is only available ...

Let $S$ be a monoid. In this paper, we prove every class of $S$-acts having a flatness property is closed underdirected colimits, it extends some known results. Furthermore thisresult implies that every $S$-act has a flatness cover if and only if it has a flatness precover.

let $s$ be a monoid. in this paper, we prove every class of $s$-acts having a flatness property is closed underdirected colimits, it extends some known results. furthermore thisresult implies that every $s$-act has a flatness cover if and only if it has a flatness precover.

let s be a monoid and x a class of s-acts which is closed under coproducts. the object of this article is to find conditions under which all s-acts have x-precovering. we have shown that the existence of torsion-free precovering implies the existence of torsion-free covering. this work is an attempt to further facilitate the study of the conjecture that all s-acts have flat cover.

In 1966 [1], Auslander introduced a class of finitely generated modules having a certain complete resolution by projective modules. Then using these modules, he defined the G-dimension (G ostensibly for Gorenstein) of finitely generated modules. It seems appropriate then to call the modules of G-dimension 0 the Gorenstein projective modules. In [4], Gorenstein projective modules (whether finite...