Volume 60 , issue 1 ( 2008 ) | back |

Centralizing and commuting generalized derivations on prime rings | 1$-$2 |

**Abstract**

Let $R$ be a prime ring and $d$ a derivation on $R$. If $f$ is a generalized derivation on $R$ such that $f$ is centralising on a left ideal $U$ of $R$, then $R$ is commutative.

**Keywords:** Prime rings; commuting and centralising maps; generalized derivations.

**MSC:** 16N60, 16S20

The resultant of non-commutative polynomials | 3$-$8 |

**Abstract**

Let $R=K[x;\sigma]$ be a skew polynomial ring over a division ring $K$. Necessary and sufficient condition under which common right factor of two skew polynomials exists is established. It is shown that the existence of common factor depends on the value of non-commutative (Dieudonné) determinant built on coefficients of polynomials and their $\sigma^{l}$-images.

**Keywords:** Polynomial ring; skew polynomial; resultant.

**MSC:** 12E15

Local Lipschitz property for the Chebyshev center mapping over $N$-nets | 9$-$22 |

**Abstract**

We prove local Lipschitz property of the map which puts in correspondence to each exact $N$-net its Chebyshev center. If dimension of Euclidean or Lobachevsky space is greater than $1$ and the net consists of more than $2$ points we show that this map is not Lipschitz in a neighbourhood of the space of all $2$-nets embedded into the space of $N$-nets endowed with Hausdorff metric.

**Keywords:** Chebyshev center; local Lipschitz property; $N$-net; Hausdorff metric.

**MSC:** 54E40, 52C35

Hypergroups of type U on the right of size five. Part two | 23$-$45 |

**Abstract**

The hypergroups $H$ of type $U$ on the right can be classified in terms of the family $P_{1}=\{1\circ x\mid x\in H\}$, where $1\in H$ is the right scalar identity. If the size of $H$ is $5$, then $P_{1}$ can assume only $6$ possible values, three of which have been studied inthe first part of the paper. In this paper, we completely describe other two of the remaining possible cases: a)~$P_{1}=\{\{1\},\{2,3\},\{4\},\{5\}\}$; b)~$P_{1}=\{\{1\},\{2,3\},\{4,5\} }$. In these cases, $P_{1}$ is a partition of $H$ and the equivalence relation associated to it is a regular equivalence on $H$. We find that, apart of isomorphisms, there are exactly $41$ hypergroups in case~a), and $56$ hypergroup in case~b).

**Keywords:** Hypergroups; hyperstructures.

**MSC:** 20N20, 05A99

Ascent and descent of weighted composition operators on $L^p$-spaces | 47$-$51 |

**Abstract**

In this paper, we study weighted composition operators on $L^p$-spaces with finite ascent and descent. We also characterize the injective weighted composition operators.

**Keywords:** Ascent; descent; measurable transformation; weighted composition operators.

**MSC:** 47B33, 46E30, 47B07, 46B70

Une note sur la noethérianité | 53$-$57 |

**Abstract**

We show that a Banach algebra in which all maximal ideals are of finite type is a noetherian algebra, hence of finite dimension. We also consider $m$-convex Fréchet algebras in which all maximal ideals are principal.

**Keywords:** Noetherian algebra; l.m.c. algebra; maximal ideal.

**MSC:** 46J10, 46J20

Hit-And-Far-Miss Topologies | 59$-$78 |

**Abstract**

We offer a unified approach to investigate hypertopologies. In this setting the proofs are simple and transparent. New problems are raised.

**Keywords:** Hypertopology, Bombay hypertopology,
hit-and-miss, hit-and-far-miss topology, locally finite, discrete,
uniformly discrete family, Hausdorff metric, Wijsman topology.

**MSC:** 54B20, 54E05, 54E15