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Title: Simple artinian rings, hereditary rings and skew fields
Author: Schofield, Aidan Harry
Awarding Body: University of London
Current Institution: Royal Holloway, University of London
Date of Award: 1984
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In this dissertation, Cohn's methods for constructing skew fields from firs are extended to methods for constructing simple artinian rings from hereditary rings; the flexibility this gives is used in order to prove results about the skew field coproduct, and other skew fields originally investigated by Cohn. Part I is devoted to developing the techniques needed; this has two main themes; the first is a detailed study of the finitely generated projective modules over an hereditary ring; the second is an investigation of the ring construction, universal localisation. The construction of universal homomorphisms from suitable hereditary rings to simple artinian rings leads to the simple artinian coproduct with amalgamation of simple artinian rings, which is the natural generalisation of the skew field coproduct of Cohn. Part II is a detailed study of the skew fields and simple artinian rings that are constructed in this way from firs or more generally from hereditary rings. The finite dimensional division subalgebras are classified (apart from the case of the skew field coproduct of two quadratic extensions of a skew field), the transcendence degree of the commutative subfields is bounded, and the centralisers of transcendental elements is studied in special cases. In addition, there are isolated results on the isomorphism classes of skew field coproducts. Part III is distinct from the rest of the dissertation; it consists of a description of generic solutions to the. partial splitting for finite dimensional central simple algebras. These are twisted forms of grassmannian varieties; this is used to show that over number fields, these varieties satisfy the Hasse principle.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: Mathematics