Greenwood, Nicholas John;
(1993)
Sigmamatrix ideals and Sigmainverting homomorphisms.
Masters thesis (M.Phil), UCL (University College London).

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Abstract
In Free Rings and their Relations, P.M.Cohn constructed a skew field from a prime matrix ideal using admissible matrices. When Σ is a lower multiplicative set of matrices over any noncommutative ring this method can be generalised to construct any epic Σinverting homomorphism upto isomorphism. This depends on the introduction of the concept of a Σmatrix ideal. Every Σinverting homomorphism gives rise to a Σmatrix ideal and conversely our main theorem shows that, given a Σmatrix ideal, an epic Σinverting homomorphism can be constructed and that the matrices which are admissible for zero are precisely those lying in the Σmatrix ideal. It is shown that the least Σmatrix ideal induces the universal Σinverting homomorphism. A description of the least Σmatrix ideal is then obtained yielding a new description of the kernel of the Σinverting homomorphism and a criterion for it to be an embedding; Malcolmson and Gerasimovs' respective descriptions of the kernel are also proved. When Σ is taken to be the complement of a prime matrix ideal the construction reduces to that used by Cohn to construct a skew field. It is further shown that the definition of a prime matrix ideal P can be simplified by restricting the class of matrices necessarily lying in P to be the hollow and degenerate matrices. The condition that P be closed with respect to row determinantal sums can be dropped completely. As a consequence, Cohn's criterion for the existence of a homomorphism from a ring to a field and the criterion for the existence of a field of fractions for a ring can be refined somewhat. For completeness Dicks and Sontag's result that Sylvester domains form the precise class of rings which have a universal field of fractions inverting all full matrices is also included.
Type:  Thesis (Masters) 

Qualification:  M.Phil 
Title:  Sigmamatrix ideals and Sigmainverting homomorphisms 
Open access status:  An open access version is available from UCL Discovery 
Language:  English 
Additional information:  Thesis digitised by ProQuest. 
Keywords:  Pure sciences; Inverting homomorphism 
URI:  https://discovery.ucl.ac.uk/id/eprint/10102045 
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