The prüfer rings that are endomorphism rings of artinian modules

In this paper we characterize the (commutative) Priifer rings that can be realized as endomorphism rings of artinian modules over arbitrary associative rings with identity (Theorem 4.7). This characterization is obtained by determining the structure of ∑-pure-injective modules over Prufer rings (Theorems 3.4 and 3.5)     

Classes of generalized ∗-modules

We study the class, STAR , of all ?-modules by means of the classes, S_λ, of all ?λ-modules, λ> 0 being a cardinal. Since STAR, equals the intersection of the decreasing chain S_λ, λ > 0, our approach ‘from above’ complements the usual approach ‘from below’ consisting in the study of quasi-progenerators and tilting modules. We present relations between categorical properties of ?-;modules and those of ?-modules. Answering a question of Menini, we use the solution of the Artin's problem to show that the chain Sλ,λ > No, is strictly decreasing in general.     

On generalized k-syzygy modules

In this paper, we first introduce the notion of generalized k-syzygy modules, and then give an equivalent characterization that the class of generalized k-syzygy modules coincides with that ofω-k-torsionfree modules. We further study the extension closure of the category consisting of generalized k-syzygy modules. Some known results are obtained as corollaries.     

Tilting modules and ∗-modules

The relation between ?-modules-studied in [MO], [D], [C], [DH], [CM], [Z] and [T]-and Tiltng modules over an arbitrary ring is analyzed. In particular we prove that Tilting modules are exactly the faithful and finendo ?-modules. This answers a question of Trlifaj [T, Problem 1.5], showing that for any ring R the class of ?-modules generating the injectives and that one of Tiltings coincides. As a first application, we give an easy proof of the fact that every faithful ?-module over a finite-dimensional K-algebra is a classical Tilting module (see [DH, Theorem 1]). As a second application, we characterise the Tiltings as those modules which induce an equivalence between two categories with suitable dual properties.     

Degeneracies of generalized inverse,vector-valued padé approximants

Wynn used generalized inverses to interpret continued fractions containing vector-valued elements. This approach led to the introduction of generalized inverse, vector-valued Padé approximants (GIPAs). All possible cases of degeneracy of GIPAs are analysed in this paper. We derive linear equations for the coefficients of the denominator polynomial of a GIPA. The solution of these equations allows construction of a GIPA in all cases where such a GIPA exists. We show that the block structure of the table of GIPAs is precisely analogous to that of the Padé table.Communicated by Edward B. Saff.     

On ∗-minimal ∗-ideals and ∗-biideals in involution rings

For semiprime involution rings, we determine some ∗-minimal ∗-ideals using idempotent elements. Nevertheless, ∗-minimal ∗-biideals are characterized by idempotent elements. Moreover, the involutive version of a theorem due to Steinfeld, which investigates a semiprime involution ring A if A=SocA, is given. Finally, semiprime involution rings having no proper nonzero ∗-biideals are characterized.     

Systems of generalized variational inequalities and their applications ∗

In this paper, we first introduce the system of generalized implicit variational inequalities and prove the existence of its solution. Then we derive existence results for systems of generalized variational and variational like inequalities and system of variational inequalities. As applications, we establish some existence results for a solution to the system of optimization problems which includes the Nash equilibrium problem as a special case     

An approach towards Schubert positivities of polynomials using Kraśkiewicz–Pragacz modules

In this paper, we investigate properties of modules introduced by Kraśkiewicz and Pragacz which realize Schubert polynomials as their characters. In particular, we give some characterizations of modules having filtrations by Kraśkiewicz–Pragacz modules. In finding criteria for such filtrations, we calculate generating sets for the annihilator ideals of the lowest vectors in Kraśkiewicz–Pragacz modules and derive a projectivity result concerning Kraśkiewicz–Pragacz modules.     

Differential étale extensions and differential modules over differential rings

This paper studies differential square zero extensions and differential modules of a commutative differential algebra R over a differential field F where the field of constants of F is algebraically closed and of characteristic 0. All elements of R and the differential R modules and algebras considered are assumed to satisfy linear homogeneous differential equations over F. For R differentially simple, we describe the invectives, and, using that all considered differential modules are R flat, provide a criterion for all square zero extensions to be differentially split.     

Group inverse and group involutory Matrices ∗

In this work we deal with group involutory matrices, i.e.A ^#=A. We give necessary and sufficient conditions to characterize these matrices in terms of different representations of the group inverse. First, we give different expressions of the group inverse of a square matrix A. In addition, the special case of integer matrices is considered.