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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
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 $ P \subset A$ the transcendence degree of $ A/P$ 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

$\displaystyle A[{X_1}, \cdots ,{X_m}]\quad {\text{and}}\quad B[{Y_1}, \cdots ,{Y_m}]$

are isomorphic, 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.

References [Enhancements On Off] (What's this?)

  • [AEH] S. Abhyankhar, P. Eakin and W. Heinzer, On the uniqueness of the coefficient ring in a ring of polynomials, J. Algebra (to appear). MR 0254023 (40:7236)
  • [CE] D. Coleman and E. Enochs, Isomorphic polynomial rings, Proc. Amer. Math. Soc. 27 (1971), 247-252. MR 0272805 (42:7686)
  • [ZS] O. Zariski and P. Samuel, Commutative algebra. Vol. 2, The University Series in Higher Math., Van Nostrand, Princeton, N.J., 1960. MR 22 #11006. MR 0120249 (22:11006)

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Keywords: Polynomial ring, noetherian ring, Krull dimension, affine ring
Article copyright: © Copyright 1972 American Mathematical Society

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