Realization problem for von Neumann regular rings

Thesis title in Czech: | Realizační problém von Neumannovsky regulárních okruhů |
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Thesis title in English: | Realization problem for von Neumann regular rings |

Key words: | von Neumannovsky regulární okruh, zjemňující monoid, |

English key words: | von Neumann regular ring, refinement monoid, |

Academic year of topic announcement: | 2015/2016 |

Type of assignment: | dissertation |

Thesis language: | angličtina |

Department: | Department of Algebra (32-KA) |

Supervisor: | doc. Mgr. Pavel Růžička, Ph.D. |

Author: | hidden - assigned and confirmed by the Study Dept. |

Date of registration: | 25.09.2015 |

Date of assignment: | 25.09.2015 |

Confirmed by Study dept. on: | 05.10.2015 |

Guidelines |

For a unital ring R, V(R) denotes the monoid of isomorphism of isomorphism classes of finitely generated projective monoids. By a result of Bergman and Dicks [9] every conical monoid with an order unit appears as V(R) of some unital hereditary ring. If a ring R is von Neumann regular, the monoid V(R) is a refinement monoid. F. Wehrung constructed a conical refinement monoid with an order unit of size aleph 2 not isomorphic to V(R) for a von Neumann regular ring [11]. Whether every refinement monoid with an order unit of a smaller size can be represented as V(R) of a regular ring remains, despite of many partial results [2,3,4], open. This problem should be the main task of the thesis. |

References |

1. G. Abrams, G. Aranda Pino, The Leavitt path algebra of a graph, J. Algebra, 293 (2005), 319-334.
2. P. Ara, The regular algebra of a poset, Trans. Amer. Math. Soc. 362 (2010), 1505-1546. 3. Pere Ara and Miquel Brustenga, The regular algebra of a quiver, J. Algebra 309 (2007), no. 1, 207–235. 4. P. Ara, K. R. Goodearl, and E. Pardo, ����₀ of purely infinite simple regular rings, ����-Theory 26 (2002), no. 1, 69–100. 5. P. Ara, K. R. Goodearl, K. C. O’Meara, and E. Pardo, Separative cancellation for projective modules over exchange rings, Israel J. Math. 105 (1998). 6. P. Ara, M. A. Moreno, and E. Pardo, Nonstable ����-theory for graph algebras, Algebr. Represent. Theory 10 (2007), no. 2, 157–178. 7. George M. Bergman, Modules over coproducts of rings, Trans. Amer. Math. Soc. 200 (1974), 1–32. 8. George M. Bergman, Coproducts and some universal ring constructions, Trans. Amer. Math. Soc. 200 (1974). 9. G. M. Bergman and W. Dicks, Universal derivations and universal ring constructions, Pacific J. Math. 79 (1978), no. 2, 293–337. 10. P. M. Cohn, Free rings and their relations, 2nd ed., London Mathematical Society Monographs, vol. 19, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1985. 11. Friedrich Wehrung, Non-measurability properties of interpolation vector spaces, Israel J. Math. 103 (1998), 177–206. |