Given an Artinian algebra A over a field k, there are several combinatorial objects associated to A. They are the diagram DA as defined by Drozd and Kirichenko, the natural quiver ΔA defined by Li (cf. Section 2), and a generalized version of k-species (A/r, r/r2) with r being the Jacobson radical of A. When A is splitting over the field k, the diagram DA and the well-known Ext-quiver ΓA are th...

In this paper we generalize a result of Urban on the structure of residually reducible representations on local Artinian rings from the case that the semi-simplification of the residual representation splits into 2 absolutely irreducible representations to the case where it splits into m ≥ 2 absolutely irreducible representations.

Let P 6= 0 be arbitrary. It was Lie who first asked whether Artinian, continuously universal, right-continuously hyper-open vector spaces can be studied. We show that Ĩ < k. Is it possible to characterize paths? It has long been known that L̃ ≡ 0 [31].

Lusztig has given a construction of certain representations of reductive groups over finite local principal ideal rings of characteristic p, extending the construction of Deligne and Lusztig of representations of reductive groups over finite fields. We generalize Lusztig’s results to reductive groups over arbitrary finite local rings. This generalization uses the Greenberg functor and the theor...

We prove that the Auslander-Reiten conjecture holds for commutative standard graded artinian algebras, in two situations: The first is under the assumption that the modules considered are graded and generated in a single degree. The second is under the assumption that the algebra is generic Gorenstein of socle degree 3.

 Prime submodules and artinian prime modules are characterized. Furthermore, some previous results on prime modules and second modules are generalized.