Let R be a right GF -closed ring with finite left and right Gorenstein global dimension. We prove that if I is an ideal of R such that R/I is a semi-simple ring, then the Gorenstein flat dimension of R/I as a right R-module and the Gorenstein injective dimension of R/I as a left R-module are identical. In particular, we show that for a simple module S over a commutative Gorenstein ring R, the G...

let r be a ring,  be an endomorphism of r and mr be a -rigid module. amodule mr is called quasi-baer if the right annihilator of a principal submodule of r isgenerated by an idempotent. it is shown that an r-module mr is a quasi-baer module if andonly if m[[x]] is a quasi-baer module over the skew power series ring r[[x; ]].

We define a notion of total acyclicity for complexes of flat quasi-coherent sheaves over a semi-separated noetherian scheme, generalising complete flat resolutions over a ring. By studying these complexes as objects of the pure derived category of flat sheaves we extend several results about totally acyclic complexes of projective modules to schemes; for example, we prove that a scheme is Goren...

We prove a generalization of the Flat Cover Conjecture by showing for any ring R that (1) each (right R-) module has a Ker Ext(−, C)-cover, for any class of pure-injective modules C, and that (2) each module has a Ker Tor(−,B)-cover, for any class of left R-modules B. For Dedekind domains, we describe Ker Ext(−, C) explicitly for any class of cotorsion modules C; in particular, we prove that (1...

Let R be an associative ring with identity. It is shown that every Σ -cotorsion left R-module satisfies the descending chain condition on divisibility formulae. If R is countable, the descending chain condition on M implies that it must be Σ -cotorsion. It follows that, for countable R, the class of Σ -cotorsion modules is closed under elementary equivalence and pure submodules. The modules M t...

is a monomorphism for all collections of right R-modules {Ni : i ∈ I} . Restricting the Ni to certain classes of modules one obtains various relativized concepts. For instance, Goodearl investigated this notion relativized to flat modules. The present study provides a uniform background for such relativizations and thus covers and generalizes, among others, a number of known results about such ...

Currently, most knowledge representation using logic programming with answer set semantics (AnsProlog) is ‘flat’. In this paper we elaborate on our thoughts about a modular structure for knowledge representation and declarative problem solving formalism using AnsProlog. We present language constructs that allow defining of modules and calling of such modules from programs. This allows one to wr...

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...