In this paper we study derived equivalences between triangular matrix algebras using certain classical recollements. We show that special properties of these recollements actually characterize triangular matrix algebras, and describe methods to construct tilting modules and tilting complexes inducing derived equivalences between them.

We prove that the cover ideals of all unimodular hypergraphs have the nonincreasing depth function property. Furthermore, we show that the index of depth stability of these ideals is bounded by the number of variables.

In this paper, we obtain a class of irreducible Virasoro modules by taking tensor products of the irreducible Virasoro modules $\Omega(\lambda,b)$ with irreducible highest weight modules $V(\theta ,h)$ or with irreducible Virasoro modules $\mathrm{Ind}_{\theta}(N)$ defined in Mazorchuk and Zhao (Selecta Math. (N.S.) 20:839–854,2014). We determine the necessary and sufficient conditions for two such irreducible tensor products to be isomorphic. Then we prove that the tensor product of $\Omega(\lambda,b)$ with a classical Whittaker module is isomorphic to the module $\mathrm{Ind}_{\theta, \lambda}(\mathbb{C}_{\mathbf{m}})$ defined in Mazorchuk and Weisner (Proc. Amer. Math. Soc. 142:3695–3703,2014). As a by-product we obtain the necessary and sufficient conditions for the module $\mathrm{Ind}_{\theta, \lambda}(\mathbb{C}_{\mathbf{m}})$ to be irreducible. We also generalize the module $\mathrm{Ind}_{\theta, \lambda}(\mathbb{C}_{\mathbf{m}})$ to $\mathrm{Ind}_{\theta,\lambda}(\mathcal{B}^{(n)}_{\mathbf{s}})$ for any non-negative integer $n$ and use the above results to completely determine when the modules $\mathrm{Ind}_{\theta,\lambda}(\mathcal{B}^{(n)}_{\mathbf{s}})$ are irreducible. The submodules of $\mathrm{Ind}_{\theta,\lambda}(\mathcal{B}^{(n)}_{\mathbf{s}})$ are studied and an open problem in Guo et al. (J. Algebra 387:68–86,2013) is solved. Feigin–Fuchs’ Theorem on singular vectors of Verma modules over the Virasoro algebra is crucial to our proofs in this paper.

Let $A = \bigoplus _{i \geqslant 0} A_i$ be a graded locally finite $k$-algebra where $A_0$ is a finite-dimensional algebra whose finitistic dimension is 0. In this paper we develop a generalized Koszul theory preserving many classical results, and show an explicit correspondence between this generalized theory and the classical theory. Applications in representations of certain categories and extension algebras of standard modules of standardly stratified algebras are described.

Let R be a ring. R is called a quasi-Frobenius (QF) ring if R is right artinian and RR is an injective right R-module. In this article, we introduce (weak) fuzzy homomorphisms of modules to obtain a fuzzy characterization of QF rings. We also obtain some fuzzy characterizations of right artinian rings and right CF rings. These results throw new light on the research of QF rings and the related CF conjecture.