Given an Artinian algebra $A$ over a field $k$, there are
several combinatorial objects associated to $A$. They are the
diagram $D_A$ as defined by Drozd and Kirichenko, the natural quiver $\Delta_A$
defined by Li (cf. Section 2), and a generalized version of
$k$-species $(A/r, r/r^2)$ with $r$ being the Jacobson radical of
$A$. When $A$ is splitting over the field $k$, the diagram $D_A$
and the well-known ext-quiver $\Gamma_A$ are the same. The main
objective of this paper is to investigate the relations among these
combinatorial objects and in turn to use these relations to give a
characterization of the algebra $A$.
The Faith-Menal conjecture is one of the three main open conjectures
on QF rings. It says that every right noetherian and left FP-injective
ring is QF. In this paper, it is proved that the conjecture is true if every
nonzero complement left ideal of the ring R is not small (or not singular).
Several known results are then obtained as corollaries.
A finite directed category is a k-linear category with finitely many objects and an underlying poset structure, where k is an algebraically closed field. This concept unifies structures such as k-linerizations of posets and finite EI categories, quotient algebras of finite-dimensional hereditary algebras, triangular matrix algebras, etc. In this paper we study representations of finite directed categories, discuss their stratification properties, and show the existence of generalized APR tilting modules for triangular matrix algebras under some assumptions.
In this paper we describe several characterizations of basic finite-dimensional k-algebras A stratified for all linear orders, and classify their graded algebras as tensor algebras satisfying some extra property. We also discuss whether for a given preorder ≼, F(≼Δ), the category of A-modules with ≼Δ-filtrations, is closed under cokernels of monomorphisms, and classify quasi-hereditary algebras satisfying this property.