It is known that a ring $R$ is left Noetherian if and only if every left $R$-module has an injective (pre)cover. We show that $\left(1\right)$ if $R$ is a right $n$-coherent ring, then every right $R$-module has an $(n,d)$-injective (pre)cover; $\left(2\right)$ if $R$ is a ring such that every $(n,0)$-injective right $R$-module is $n$-pure extending, and if every right $R$-module has an $(n,0)$-injective cover, then $R$ is right $n$-coherent. As applications of these results, we give some characterizations of $(n,d)$-rings, von Neumann regular rings and semisimple rings....

Based on the work of D. Happel, I. Reiten and S. Smalø on quasitilted artin algebras, the first two authors recently introduced the notion of quasitilted rings. Various authors have presented examples of quasitilted artin algebras that are not tilted. Here we present a class of right quasitilted rings that not right tilted, and we show that they satisfy a condition that would force a quasitilted artin algebra to be tilted.

We provide a characterization of artin algebras without chains of nonzero homomorphisms between indecomposable finitely generated modules starting with an injective module and ending with a projective module.

We discuss the existence of tilting modules which are direct limits of finitely generated tilting modules over tilted algebras.

This paper contains the algebraic analog for idempotent matrices of the Chern-Weil theory of characteristic classes. This is used to show, algebraically, that the canonical line bundle on the complex projective space is not stably trivial. Also a theorem is proved saying that for any smooth manifold there is a canonical epimorphism from the even dimensional algebraic de Rham cohomology of its algebra of smooth functions onto the standard even dimensional de Rham cohomology of the manifold.

The aim of this work is to characterize the algebras which are standardly stratified with respect to any order of the simple modules. We show that such algebras are exactly the algebras with all idempotent ideals projective. We also deduce as a corollary a characterization of hereditary algebras, originally due to Dlab and Ringel.

We present general properties for almost-flat modules and we prove that a self-small right module is almost flat as a left module over its endomorphism ring if and only if the class of $g$-static modules is closed under the kernels.

We give axiomatic conditions in order to calculate the local cohomology of some idempotent kernel functors. These results lie in some new dimension introduced by T. Levasseur for Auslander-Gorenstein rings. Under some hypothesis, we generalize previous results.

We complete the derived equivalence classification of all weakly symmetric algebras of domestic type over an algebraically closed field, by solving the problem of distinguishing standard and nonstandard algebras up to stable equivalence, and hence derived equivalence. As a consequence, a complete stable equivalence classification of weakly symmetric algebras of domestic type is obtained.

In this note we show that for a ${*}^n$-module, in particular, an almost $n$-tilting module, $P$ over a ring $R$ with $A={\mathrm{E}nd}_{R}P$ such that $P_A$ has finite flat dimension, the upper bound of the global dimension of $A$ can be estimated by the global dimension of $R$ and hence generalize the corresponding results in tilting theory and the ones in the theory of $*$-modules. As an application, we show that for a finitely generated projective module over a VN regular ring $R$, the global dimension of its endomorphism ring is not more...

To a commutative ring K, and a family of K-algebras indexed by the vertex set of a graph, we associate a K-algebra obtained by a mixture of coproduct and tensor product constructions. For this, and related constructions, we give exact sequences and deduce homological properties.

Using derived categories, we develop an alternative approach to defining Koszulness for positively graded algebras where the degree zero part is not necessarily semisimple.

Let $R$ be a ring. A left $R$-module $M$ is called an FC-module if $M^{+}={Hom}_{\mathbb{Z}}(M,\mathbb{Q}/\mathbb{Z})$ is a flat right $R$-module. In this paper, some homological properties of FC-modules are given. Let $n$ be a nonnegative integer and ${\mathrm{ℱ𝒞}}_n$ the class of all left $R$-modules $M$ such that the flat dimension of $M^{+}$ is less than or equal to $n$. It is shown that $({}^{\perp }\left({\mathrm{ℱ𝒞}}_n^{\perp }\right),{\mathrm{ℱ𝒞}}_n^{\perp })$ is a complete cotorsion pair and if $R$ is a ring such that $fd\left({\left({}_{R}R\right)}^{+}\right)\le n$ and ${\mathrm{ℱ𝒞}}_n$ is closed under direct sums, then $({\mathrm{ℱ𝒞}}_n,{\mathrm{ℱ𝒞}}_n^{\perp })$ is a perfect cotorsion pair. In particular, some known results are obtained as corollaries....