let r be an associative ring with identity, c(r) be the category of com-plexes of r-modules and flat(c(r)) be the class of all at complexes of r-modules. we show that the at cotorsion theory (flat(c(r)); flat(c(r))−)have enough injectives in c(r). as an application, we prove that for each atcomplex f and each complex y of r-modules, exti (f,x)= 0, whenever ris n-perfect and i > n.

Let R be a subring of the rationals. We want to investigate self splitting R-modules G that is ExtR(G,G) = 0 holds and follow Schultz [22] to call such modules splitters. Free modules and torsion-free cotorsion modules are classical examples for splitters. Are there others? Answering an open problem by Schultz [22] we will show that there are more splitters, in fact we are able to prescribe the...

Rooted quivers are quivers that do not contain A∞ ≡ ··· → • → • as a subquiver. The existence of flat covers and cotorsion envelopes for representations of these quivers have been studied by Enochs et al. The main goal of this paper is to prove that flat covers and cotorsion envelopes exist for representations of A∞. We first characterize finitely generated projective representations of A∞. We ...

Let $\mathscr{C}$ be an extriangulated category with a proper class $\xi$ of $\mathbb{E}$-triangles. We introduce the notions left Frobenius pairs, ($n$-)cotorsion pairs and (weak) Auslander-Buchweitz contexts respect to in $\mathscr{C}$. show how construct cotorsion pais from $n$-cotorsion establish one-to-one correspondence between contexts. also study relation certain that pairs. These work ...

Abstract We show that a direct limit of projective contramodules (over right linear topological ring) is if it has cover. A similar result obtained for $\infty $-strictly flat dimension not exceeding $1$, using an argument based on the notion Jacobson radical. Covers and precovers limits more general classes objects, both in abelian categories with exact nonexact limits, are also discussed, eye...

In this paper, the notion of cotorsion classes is introduced into higher homological algebra . Our results motivate definition, showing that n -cotorsion satisfies usual properties one could expect. particular, a version Wakamatsu's Lemma proved. At last section, connections with wide subcategories are studied.

We explore the interlacing between model category structures attained to classes of modules of finite X -dimension, for certain classes of modules X . As an application we give a model structure approach to the Finitistic Dimension Conjectures and present a new conceptual framework in which these conjectures can be studied. Let Λ be a finite dimensional algebra over a field k (or more generally...

Let R be a ring and T be a 1-tilting right R-module. Then T is of countable type. Moreover, T is of finite type in case R is a Prüfer domain.