Matrix localizations of 𝑛-firs. I

Malcolmson, Peter

doi:10.1090/S0002-9947-1984-99925-9

Matrix localizations of $n$-firs. I
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by Peter Malcolmson

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

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

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

Abstract:

An $n$-fir is an associative ring in which every $n$-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 $n$-fir with $n > 2k$ and if we form a matrix localization by adjoining universal inverses to all full matrices of size $k$, then the resulting ring is an $(n - 2k)$-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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