Quotients of valuated vector spaces

Author:
Paul Hill

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

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

MathSciNet review:
589128

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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****[2]**László Fuchs,*Subfree valued vector spaces*, Abelian group theory (Proc. Second New Mexico State Univ. Conf., Las Cruces, N.M., 1976) Springer, Berlin, 1977, pp. 158–167. Lecture Notes in Math., Vol. 616. MR**0480700****[3]**Paul Hill,*Criteria for freeness in groups and valuated vector spaces*, Abelian group theory (Proc. Second New Mexico State Univ. Conf., Las Cruces, N.M., 1976) Springer, Berlin, 1977, pp. 140–157. Lecture Notes in Math., Vol. 616. MR**0486206****[4]**Paul Hill and Errin White,*The projective dimension of valuated vector spaces*, J. Algebra**74**(1982), no. 2, 374–401. MR**647246**, 10.1016/0021-8693(82)90031-X**[5]**Fred Richman and Elbert A. Walker,*Valuated groups*, J. Algebra**56**(1979), no. 1, 145–167. MR**527162**, 10.1016/0021-8693(79)90330-2**[6]**Errin Erb White,*On the homological dimension of valuated vector spaces*, Canad. Math. Bull.**24**(1981), no. 1, 97–100. MR**611216**, 10.4153/CMB-1981-015-4

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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