A standard associative graded algebra R over a field k is called Koszul if k admits a linear resolution as an R-module. A (right) R-module M is called Koszul if it admits a linear resolution too. Here we study a special class of Koszul algebras — roughly say, algebras having a lot of Koszul cyclic modules. Commutative algebras with similar properties (so-called algebras with Koszul filtrations)...

In this article, we formalize some basic facts of Z-module. In the first section, we discuss the rank of submodule of Z-module and its properties. Especially, we formally prove that the rank of any Z-module is equal to or more than that of its submodules, and vice versa, there exists a submodule with any given rank that satisfies the above condition. In the next section, we mention basic facts ...

This article concerns linear parts of minimal resolutions of finitely generated modules over commutative local, or graded rings. The focus is on the linearity defect of a module, which marks the point after which the linear part of its minimal resolution is acyclic. The results established track the change in this invariant under some standard operations in commutative algebra. As one of the ap...

Let R be a noetherian ring and M a finite R-module. With a linear form χ on M one associates the Koszul complex K(χ). If M is a free module, then the homology of K(χ) is well-understood, and in particular it is grade sensitive with respect to Imχ. In this note we investigate the case of a module M of projective dimension 1 (more precisely, M has a free resolution of length 1) for which the firs...

Let R be a noetherian ring andM a finite R-module. With a linear form χ onM one associates the Koszul complex K(χ). IfM is a free module, then the homology of K(χ) is well-understood, and in particular it is grade sensitive with respect to Imχ. In this note we investigate the case of a module M of projective dimension 1 (more precisely, M has a free resolution of length 1) for which the first n...

A process of computation in a two-dimensional mesh of processors is analysedd in this work. A computational taskk is assumedd to be arbitrarily divisible between processors. The loadd of the job is transferre dd from the originatingg processor to other processors by a point to point communication n etwork. Communication delays are taken into consideration. The time costs of computation andd com...

SUMMARY In this work, we present a CUDA-based GPU implementation of a Poisson-Boltzmann equation solver, in both the linear and non-linear versions, using double precision. A finite difference scheme is adopted and made suitable for the GPU architecture. The resulting code was interfaced with the electrostatics software for biomolecules DelPhi, which is widely used in the computational biology ...

Minimality ranks in the style of Deissler are one way of measuring the structural complexity of minimal extensions of first-order structures. In particular, positive Deissler rank measures the complexity of the injective envelope of a module as an extension of that module. In this paper we solve a problem of the second author by showing that certain injective envelopes have the maximum possible...

The characterization of H-prime radical is given in many ways. Meantime, the relations between the radical of smash product R#H and the H-radical of Hopf module algebra R are obtained. 0 Introduction and Preliminaries In this paper, let k be a commutative associative ring with unit, H be an algebra with unit and comultiplication △ ( i.e. ∆ is a linear map: H → H ⊗H), R be an algebra over k (R m...

Let 2 be a polyhedral complex embedded in the euclidean space Ed and Sr(2), r ≥ 0, denote the set of all Cr-splines on 2. Then Sr(2) is an R-module where R = E[x1, . . . , xd] is the ring of polynomials in several variables. In this paper we state and prove the existence of an algorithm to write down a free basis for the above R-module in terms of obvious linear forms defining common faces of m...