A note on the uniqueness of rings of coefficients in polynomial rings

Authors:
Paul Eakin and K. K. Kubota

Journal:
Proc. Amer. Math. Soc. **32** (1972), 333-341

MSC:
Primary 13B25

DOI:
https://doi.org/10.1090/S0002-9939-1972-0297763-8

MathSciNet review:
0297763

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Abstract: We say that the ring *A* is of transcendence degree *n* over its subfield *k* if for every prime the transcendence degree of over *k* is at most *n* and equality is attained for some *P*. In this paper we prove the following: Suppose *A* is a noetherian ring of transcendence degree one over its subfield *k*. Then if *B* is any ring such that the polynomial rings

*A*is isomorphic to

*B*. Moreover if

*A*has no nontrivial idempotents then either

*A*is isomorphic to the polynomials in one variable over a local artinian ring or, modulo the nil radical, the given isomorphism takes

*A*onto

*B*.

**[AEH]**Paul Eakin and William Heinzer,*Non finiteness in finite dimensional Krull domains*, J. Algebra**14**(1970), 333–340. MR**0254023**, https://doi.org/10.1016/0021-8693(70)90109-2**[CE]**D. B. Coleman and E. E. Enochs,*Isomorphic polynomial rings*, Proc. Amer. Math. Soc.**27**(1971), 247–252. MR**0272805**, https://doi.org/10.1090/S0002-9939-1971-0272805-3**[ZS]**Oscar Zariski and Pierre Samuel,*Commutative algebra. Vol. II*, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N. J.-Toronto-London-New York, 1960. MR**0120249**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1972-0297763-8

Keywords:
Polynomial ring,
noetherian ring,
Krull dimension,
affine ring

Article copyright:
© Copyright 1972
American Mathematical Society