## Perspectives in Ring TheoryFreddy Van Oystaeyen, Lieven le Bruyn This proceedings is composed of the papers resulting from the NATO work-shop "Perspectives in Ring Theory" and the work-shop "Geometry and Invariant The ory of Representations of Quivers" . Three reports on problem sessions have been induced in the part corresponding to the work-shop where they belonged. One more report on a problem session, the "lost" problem session, will be published elsewhere eventually. vii Acknowledgement The meeting became possible by the financial support of the Scientific Affairs Division of NATO. The people at this division have been very helpful in the orga nization of the meeting, in particular we commemorate Dr. Mario di Lullo, who died unexpectedly last year, but who has been very helpful with the organization of earlier meetings in Ring Theory. For additional financial support we thank the national foundation for scientific research (NFWO), the rector of the University of Antwerp, UIA, and the Belgian Ministry of Education. We also gladly acknowledge support from the Belgian Friends of the Hebrew University and the chairman Prof. P. Van Remoortere who honored Prof. S. Amitsur for his continuous contributions to the mathematical activities at the University of Antwerp. I thank the authors who contributed their paper(s) to this proceedings and the lecturers for their undisposable contributions towards the success of the work-shop. Finally I thank Danielle for allowing me to spoil another holiday period in favor of a congress. |

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### Contents

Le Bruyn A Schofield Rational Invariants of Quivers | 21 |

Y Teranishi A Theorem on Invariants of Semisimple | 37 |

Schaps compilation Report and Open Questions from a Workshop | 59 |

S A Amitsur An Example in Central Division Algebras 85 | 93 |

Childs On Hopf Galois Extensions of Fields | 117 |

T J Ford On the Brauer Group and the Cup Product Map 135 | 153 |

A Leroy J Matczuk On the GelfandKirillov Dimension | 204 |

J Matczuk The GelfandKirillov Dimension | 221 |

S Montgomery Progress on Some Problems about | 269 |

J Okminski A Note on the P I Property | 275 |

Liu Shaoxue F Van Oystaeyen Group Graded Rings Smash Products | 299 |

H Tachikawa Reflexive AuslanderReiten Sequences | 311 |

Y Teranishi Noncommutative Invariant Theory | 321 |

Van den Bergh The BrauerSeveri Scheme of the Trace | 332 |

F Van Oystaeyen Some Problems on Associative Rings | 339 |

A R Wadsworth Dubrovin Valuation Rings | 359 |

McConnell J C Robson The Nulstellensatz and Generic Flatness | 227 |

McConnell J C Robson GelfandKirillov Dimension | 233 |

Menini GabrielPopescu Type Theorems | 239 |

Zacharia Special Monomial Algebras | 375 |

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1988 by Kluwer A-module abelian action affine Artinian assume automorphism Azumaya algebra basis Brauer group Bruyn eds Bsevn(A central simple algebra cocycle cohomology consider Corollary corresponding crossed product defined deformation denote division algebra division ring Dubrovin valuation ring elements endomorphism enveloping algebra equivalent exact sequence example finite dimensional finite group finite module functor Galois extensions GK.dim A[t graded ring group G hence homomorphism Hopf algebra idempotents induces invariant field isomorphic k-algebra Kluwer Academic Publishers lattice Lemma Lie algebra linear Math matrices maximal minimal polynomial monic monomial morphism multiplication Noetherian noncommutative Oystaeyen Perspectives in Ring polynomial ring prime ideals problem projective Proof Proposition prove quiver Q quotient rank rational invariants representation result right invariant right semi-invariant polynomials Ring Theory Schur root smash product stably isomorphic subgroup subring surjective Theorem tilting module vector space zero