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

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- by Paul Eakin and K. K. Kubota
- Proc. Amer. Math. Soc.
**32**(1972), 333-341 - DOI: https://doi.org/10.1090/S0002-9939-1972-0297763-8
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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 \[ A[{X_1}, \cdots ,{X_m}]\quad {\text {and}}\quad B[{Y_1}, \cdots ,{Y_m}]\] 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

- Paul Eakin and William Heinzer,
*Non finiteness in finite dimensional Krull domains*, J. Algebra**14**(1970), 333–340. MR**254023**, DOI 10.1016/0021-8693(70)90109-2 - D. B. Coleman and E. E. Enochs,
*Isomorphic polynomial rings*, Proc. Amer. Math. Soc.**27**(1971), 247–252. MR**272805**, DOI 10.1090/S0002-9939-1971-0272805-3 - 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**

## Bibliographic Information

- © Copyright 1972 American Mathematical Society
- 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