Algebraic extensions of power series rings

Author:
Jimmy T. Arnold

Journal:
Trans. Amer. Math. Soc. **267** (1981), 95-110

MSC:
Primary 13J05; Secondary 13G05

DOI:
https://doi.org/10.1090/S0002-9947-1981-0621975-4

MathSciNet review:
621975

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Abstract: Let and be integral domains such that and is not algebraic over . Is it necessarily the case that there exists an integral domain such that and ? While the general question remains open, the question is answered affirmatively in a number of cases. For example, if satisfies any one of the conditions (1) is Noetherian, (2) is integrally closed, (3) the quotient field of is countably generated as a ring over , or (4) has Krull dimension one, then an affirmative answer is given. Further, in the Noetherian case it is shown that is algebraic over if and only if it is integral over and necessary and sufficient conditions are given on and in order that this occur. Finally if, for every positive integer , implies that , then it is shown that is algebraic over for every .

**[1]**J. T. Arnold and D. W. Boyd,*Transcendence degree in power series rings*, J. Algebra**57**(1979), no. 1, 180–195. MR**533108**, https://doi.org/10.1016/0021-8693(79)90216-3**[2]**Robert Gilmer,*A note on the quotient field of the domain 𝐷[[𝑋]]*, Proc. Amer. Math. Soc.**18**(1967), 1138–1140. MR**0217060**, https://doi.org/10.1090/S0002-9939-1967-0217060-4**[3]**Robert Gilmer,*Integral dependence in power series rings*, J. Algebra**11**(1969), 488–502. MR**0234950**, https://doi.org/10.1016/0021-8693(69)90088-X**[4]**Robert Gilmer,*Multiplicative ideal theory*, Marcel Dekker, Inc., New York, 1972. Pure and Applied Mathematics, No. 12. MR**0427289****[5]**P. B. Sheldon,*How changing**changes its quotient field*, Trans. Amer. Math. Soc.**122**(1966), 321-333.**[6]**B. L. Van der Waerden,*Modern algebra*. I, Ungar, New York, 1949.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1981-0621975-4

Keywords:
Power series ring,
algebraic extension,
integral extension

Article copyright:
© Copyright 1981
American Mathematical Society