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

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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]**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)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
13B25

Retrieve articles in all journals with MSC: 13B25

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