Quotients of valuated vector spaces

Author:
Paul Hill

Journal:
Proc. Amer. Math. Soc. **81** (1981), 14-18

MSC:
Primary 18G05; Secondary 15A03, 18G20, 20K10

DOI:
https://doi.org/10.1090/S0002-9939-1981-0589128-1

MathSciNet review:
589128

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Abstract | References | Similar Articles | Additional Information

Abstract: This paper has three purposes. The first is to present a direct example of a quotient of an injective space that is not itself injective. The second is to demonstrate that Theorem 3 in [**2**] is false. An -dense subspace of a free space is not free by virtue of having only values that are cofinal with . Finally, we furnish a valid proof, independent of the aforementioned Theorem 3, that a subspace of a free space is free if satisfies Fuchs' countability condition.

**[1]**L Fuchs,*Vector spaces with valuations*, J. Algebra**35**(1975), 23-38. MR**0371995 (51:8212)****[2]**-,*Subfree valued vector spaces*, Lecture Notes in Math., Vol. 616, Springer-Verlag, Berlin and New York, 1977, pp. 158-167. MR**0480700 (58:854)****[3]**P. Hill,*Criteria for freeness in groups and valuated vector spaces*, Lecture Notes in Math., Vol. 616, Springer-Verlag, Berlin and New York, 1977, pp. 140-157. MR**0486206 (58:5978)****[4]**P. Hill and E. White,*The projective dimension of valuated vector spaces*(preprint). MR**647246 (84d:18014)****[5]**F. Richman and E. A. Walker,*Valuated groups*, J. Algebra**56**(1979), 145-167. MR**527162 (80k:20053)****[6]**E. White,*On the homological dimension of valuated vector spaces*, Canad. Bull. Math. (to appear). MR**611216 (82d:20054)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1981-0589128-1

Keywords:
Valuation,
valuated vector space,
free space,
quotient of injective,
-dense subspace,
separable subspace,
projective dimension greater than one,
criterion for freeness

Article copyright:
© Copyright 1981
American Mathematical Society