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From Hermite rings to Sylvester domains
Author(s):
P.
M.
Cohn
Journal:
Proc. Amer. Math. Soc.
128
(2000),
1899-1904.
MSC (1991):
Primary 16E60;
Secondary 15A30, 16D40
Posted:
November 1, 1999
MathSciNet review:
1646314
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Abstract:
The main result proved here is a new criterion for a ring to be a Sylvester domain, and so to have a universal skew field of fractions inverting all full matrices: An Hermite ring is a Sylvester domain if and only if any product of full matrices (when defined) is full. This is also shown to hold if (and only if) the set of all full matrices is lower multiplicative. The definition of Hermite rings is weakened, but it is shown that in any case infinitely many sentences are needed.
References:
- [1]
- P. M. Cohn, Some remarks on the invariant basis property, Topology 5 (1966), 215-228. MR 33:5676
- [2]
- P. M. Cohn, Un critère d'immersibilité d'un anneau dans un corps gauche, Comptes Rendus Acad. Sci. (Paris) Sér. A 272 (1971), 1442-1444.
- [3]
- P. M. Cohn, The class of rings embeddable in skew fields, Bull. London Math. Soc., 6 (1974), 147-148.MR 51:3216
- [4]
- P. M. Cohn, Universal Algebra (revised and enlarged edition), D. Reidel, Dordrecht-Boston, 1981.
- [5]
- P. M. Cohn, Free Rings and their Relations, 2nd edition, LMS Monographs No. 19, Academic Press, London and Orlando, 1985. MR 87e:16006
- [6]
- P. M. Cohn, Skew Fields, Theory of General Division Rings, Encyclopedia of Mathematics and its Applications, Vol. 57, Cambridge University Press, 1995. MR 97d:12003
- [7]
- P. M. Cohn and A. H. Schofield, On the law of nullity, Math. Proc. Camb. Phil. Soc. 91 (1982), 357-374.MR 83h:16004
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Additional Information:
P.
M.
Cohn
Affiliation:
University College London, Gower Street, London WC1E 6BT, United Kingdom
Email:
pmc@math.ucl.ac.uk
DOI:
10.1090/S0002-9939-99-05189-8
PII:
S 0002-9939(99)05189-8
Received by editor(s):
April 17, 1998
Received by editor(s) in revised form:
August 24, 1998
Posted:
November 1, 1999
Communicated by:
Ken Goodearl
Copyright of article:
Copyright
2000,
American Mathematical Society
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