de Oliveira Ferreira, Vitor; (1999) Firs under field extensions. Doctoral thesis (Ph.D), UCL (University College London).

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Abstract

Our main aim is to investigate the relation between free ideal rings (firs) which are algebras over a field and the rings obtained from them by a commutative field extension of scalars. Having in mind the monoid of projectives of these rings, we prove that commutative monoids with distinguished element which are conical and have the UGN property are strongly embedded in their coproduct and that the coproduct inherits some properties of its factors. This result, in conjunction with Bergman's coproduct theorems, is used to establish links between coproducts of skew fields and the rings obtained from them by extension of scalars. The notion of a power-free ideal ring is explored when looking at coproducts and, more generally, at rings obtained by matrix reduction of coproducts. We also look at firs of the form R = Fk(X), where F is either a finite Galois extension or a simple purely inseparable extension of k. These firs, when tensored with F over k, give rise to rings that are no longer firs, but they are very close to being full matrix rings over firs in the sense that the adjunction to R of a single inverse (in the purely inseparable case) or of finitely many inverses (in the Galois case) originates a ring which under the same extension of scalars is a full matrix ring over a fir. The universal field of fractions of a fir obtained by this construction is just the skew field component of the simple artinian ring obtained by extending the scalars of the universal field of fractions of R.

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