Monomial curves and Cohen-Macaulayness of their tangent cones


Feza Arslan

Monomial curves curves in affine l-space are curves having parameterization

x1=tn1, x2=tn2, ¼, xl=tnl
. In the first part, we give some general results and open problems about monomial curves. In the second part, we talk how Grobner theory can be used to check the Cohen-Macalayness of tangent cones of monomial curves.

A Remark on the Minimal Polynomial of the Product of Linear Recurring Sequences

A Remark on the Minimal Polynomial of the Product of Linear Recurring Sequences


Emrah Çakçak
Department of Mathematics, Middle East Technical University,
06531 Ankara, Turkey
E-mail: cakcak@rorqual.cc.metu.edu.tr

1

Let F be an arbitrary field. For monic polynomials f,g Î F[x] let M\sb F(f) and M\sb F(g) be the set of all linear recurring sequences in F with minimal polynomial f and g, respectively. Our result concerns the minimal polynomial of the product sequence st = (s\sb nt\sb n)\sp¥\sb n=0 with s = (s\sb n)\sp ¥\sb n=0 Î M\sb F(f) and t = (t\sb n)\sp ¥\sb n=0 Î M\sb F(g). We use the techniques given by R. Göttfert and H. Niederreiter in the paper ''On the Minimal Polynomial of the Product of Linear Recurring Sequences'' [Finite Fields Appl. 1 (1995), no. 2, 204-218], to give a better lower bound on the linear complexity of st with s Î M\sb F(f) and t Î M\sb F(g). The result is an improvement of that of R. Göttfert and H. Niederreiter.

The $p$-Cockcroft property of central extensions of groups

The p-Cockcroft property of central extensions of groups


A. Sinan Çevik
Balikesir Universitesi, Fen-Edebiyat Fakultesi
Matematik Bolumu, 10100 Balikesir/Turkey

A presentation for an arbitrary group extension is well-known, see for instance [BHP]. Also a generalization of the work in [CCS] on central extensions is presented in [BHP]. As an application of this we discuss necessary and sufficient conditions for the presentation of the central extension to be p-Cockcroft, where p is a prime or 0. Finally, we present some examples of this result.
1991 Mathematics Subject Classification: 20F05, 20F55, 20F32, 57M05, 57M20.


    [BHP ] Y.G. Baik, J. Harlander and S. J. Pride, The geometry of group extensions, to be appear in Journal of Group Theory.
    [CCS ] J.H. Conway, H.S.M. Coxeter, G.C.Shephard, The center of a finitely generated group, Tensor. 25 (1972), 405-418.

Commutative Rings $R$ with $\z*(R)=0$

Commutative Rings R with Z**(R)=0


A. Çi~gdem Özcan
Let R be a ring with identity and M be a right R-module. E(M) is the injective hull of M. M is called a small module if whenever M+N=E(M) for some submodule N of E(M) we have N=E(M). We set
Z**(M)={ m Î M : mR is small }.
Z**(M) is a submodule of M. It is known that if every simple right R-module is injective (i.e. R is a right V-ring) then Z**(RR)=0. Over a commutative ring R, R is a V-ring if and only if R is a (von Neumann) regular ring. In the light of these results we study on the following question:

Is any commutative ring R with Z**(R)=0 a regular ring?

We give an example of a commutative ring R with Z**(R)=0 but R is not a regular ring. So the answer to the question is no. We also prove the following theorems.

Theorem 1. Let R be a commutative ring with Z**(R)=0. If R has finite uniform dimension then R is semiprime Artinian.

Theorem 2. R is a semiprime Artinian ring if and only if R is a right fully bounded ring with only a finite number of minimal prime ideals such that R has finite uniform dimension and Z**(RR)=0.

Footnotes:

1This work is a part of my M.Sc. Thesis which is carried out under the guidance of Prof. Ersan Aky ld z.

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