Matrix localizations of -firs. I

Author:
Peter Malcolmson

Journal:
Trans. Amer. Math. Soc. **282** (1984), 503-518

MSC:
Primary 16A06; Secondary 16A08

Part II:
Trans. Amer. Math. Soc. (2) (1984), 519--527

MathSciNet review:
732103

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Abstract: An -fir is an associative ring in which every -generator right ideal is free of unique rank. Matrix localization of a ring involves the adjunction of universal inverses to certain matrices over the ring, so that a new ring results over which the matrices have inverses, but so that the minimum of additional relations is imposed. A full matrix is a square matrix which, when considered as an endomorphism of a free module, cannot be factored through a free module of smaller rank. The main result of this paper is that if the original ring is an -fir with and if we form a matrix localization by adjoining universal inverses to all full matrices of size , then the resulting ring is an -fir. This generalizes an announced result of V. N. Gerasimov. There are related results on the structure of the universal skew field of fractions of a semifir.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1984-99925-9

Keywords:
Matrix localization,
-fir,
universal localization

Article copyright:
© Copyright 1984
American Mathematical Society